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<head><title>Matrix -- the class of all matrices</title>
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<div><h1>Matrix -- the class of all matrices</h1>
<div class="single"><h2>Description</h2>
<div>A matrix is a homomorphism between two modules, together with an integer (or vector of integers) called its degree, which is used when determining whether the map is homogeneous. The matrix is stored in the usual way as a rectangular array of ring elements. When the source or target modules are not free, the matrix is interpreted as a linear transformation in terms of the generators of the modules.<p/>
A matrix <tt>f</tt> is an immutable object, so if you want to cache information about it, put it in the hash table <tt>f.cache</tt>.<p/>
Common ways to make a matrix:<ul><li><span><a href="_map.html" title="make a map">map</a> -- make a map</span></li>
<li><span><a href="_matrix.html" title="make a matrix">matrix</a> -- make a matrix</span></li>
</ul>
Common ways to get information about matrices:<ul><li><span><a href="_degree_lp__Matrix_rp.html" title="">degree(Matrix)</a></span></li>
<li><span><a href="_is__Homogeneous.html" title="whether something is homogeneous (graded)">isHomogeneous(Matrix)</a> -- whether something is homogeneous (graded)</span></li>
<li><span><a href="_matrix_lp__Matrix_rp.html" title="the matrix between generators">matrix(Matrix)</a> -- the matrix between generators</span></li>
</ul>
Common operations on matrices:<ul><li><span><a href="__pl.html" title="a unary or binary operator, usually used for addition">Matrix + Matrix</a> -- a unary or binary operator, usually used for addition</span></li>
<li><span><a href="_-.html" title="a unary or binary operator, usually used for negation or subtraction">Matrix - Matrix</a> -- a unary or binary operator, usually used for negation or subtraction</span></li>
<li><span><a href="__st.html" title="a binary operator, usually used for multiplication">RingElement * Matrix</a> -- a binary operator, usually used for multiplication</span></li>
<li><span><a href="___Matrix_sp_st_sp__Matrix.html" title="matrix multiplication">Matrix * Matrix</a> -- matrix multiplication</span></li>
<li><span><a href="__eq_eq.html" title="equality">Matrix == Matrix</a> -- equality</span></li>
<li><span><a href="___Matrix_sp_pl_pl_sp__Matrix.html" title="direct sum of maps">Matrix ++ Matrix</a> -- direct sum of maps</span></li>
<li><span><a href="___Matrix_sp_st_st_sp__Matrix.html" title="tensor product">Matrix ** Matrix</a> -- tensor product</span></li>
<li><span><a href="_methods_spfor_spnormal_spforms_spand_spremainder.html" title="calculate the normal form of ring elements and matrices">Matrix % Matrix</a> -- calculate the normal form of ring elements and matrices</span></li>
<li><span><a href="___Matrix_sp_sl_sl_sp__Matrix.html" title="factor a map through another">Matrix // Matrix</a> -- factor a map through another</span></li>
<li><span><a href="___Matrix_sp_vb_sp__Matrix.html" title="join matrices horizontally">Matrix | Matrix</a> -- join matrices horizontally</span></li>
<li><span><a href="___Matrix_sp_vb_vb_sp__Matrix.html" title="join matrices vertically">Matrix || Matrix</a> -- join matrices vertically</span></li>
<li><span><a href="___Matrix_sp^_sp__List.html" title="select rows">Matrix ^ List</a> -- select rows</span></li>
<li><span><a href="___Matrix_sp_us_sp__List.html" title="select columns">Matrix _ List</a> -- select columns</span></li>
</ul>
Common ways to use a matrix:<ul><li><span><a href="_cokernel.html" title="cokernel of a map of modules, graded modules, or chaincomplexes">cokernel(Matrix)</a> -- cokernel of a map of modules, graded modules, or chaincomplexes</span></li>
<li><span><a href="_image.html" title="image of a map">image(Matrix)</a> -- image of a map</span></li>
<li><span><a href="_kernel_lp__Matrix_rp.html" title="kernel of a matrix">kernel(Matrix)</a> -- kernel of a matrix</span></li>
<li><span><a href="_homology_lp__Matrix_cm__Matrix_rp.html" title="homology of a pair of maps">homology(Matrix,Matrix)</a> -- homology of a pair of maps</span></li>
</ul>
</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_matrices.html" title="">matrices</a></span></li>
</ul>
</div>
<div class="waystouse"><h2>Functions and methods returning a matrix :</h2>
<ul><li><span>Matrix + Matrix, see <span><a href="__pl.html" title="a unary or binary operator, usually used for addition">+</a> -- a unary or binary operator, usually used for addition</span></span></li>
<li><span>- Matrix, see <span><a href="_-.html" title="a unary or binary operator, usually used for negation or subtraction">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li>
<li><span>Matrix - Matrix, see <span><a href="_-.html" title="a unary or binary operator, usually used for negation or subtraction">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li>
<li><span><a href="_adjoint_lp__Matrix_cm__Module_cm__Module_rp.html" title="an adjoint map">adjoint(Matrix,Module,Module)</a> -- an adjoint map</span></li>
<li><span><a href="_adjoint1_lp__Matrix_cm__Module_cm__Module_rp.html" title="an adjoint map">adjoint1(Matrix,Module,Module)</a> -- an adjoint map</span></li>
<li><span><a href="_ambient_lp__Matrix_rp.html" title="">ambient(Matrix)</a></span></li>
<li><span><a href="_basis.html" title="basis of all or part of a module or ring">basis</a> -- basis of all or part of a module or ring</span></li>
<li><span><a href="_borel_lp__Matrix_rp.html" title="make a Borel fixed submodule">borel(Matrix)</a> -- make a Borel fixed submodule</span></li>
<li><span><a href="___Chain__Complex__Map_sp_us_sp__Z__Z.html" title="component map">ChainComplexMap _ ZZ</a> -- component map</span></li>
<li><span>GradedModuleMap _ ZZ, see <span><a href="___Chain__Complex__Map_sp_us_sp__Z__Z.html" title="component map">ChainComplexMap _ ZZ</a> -- component map</span></span></li>
<li><span><a href="_complement_lp__Matrix_rp.html" title="find the minimal generators for cokernel of a matrix (low level form)">complement(Matrix)</a> -- find the minimal generators for cokernel of a matrix (low level form)</span></li>
<li><span>compress(Matrix), see <span><a href="_compress.html" title="extract nonzero columns from a matrix">compress</a> -- extract nonzero columns from a matrix</span></span></li>
<li><span><a href="_contract_sq_lp__Matrix_cm__Matrix_rp.html" title="contract a matrix by a matrix, the dual notion">contract'(Matrix,Matrix)</a> -- contract a matrix by a matrix, the dual notion</span></li>
<li><span><a href="_contract_lp__Matrix_cm__Matrix_rp.html" title="contract a matrix by a matrix">contract(Matrix,Matrix)</a> -- contract a matrix by a matrix</span></li>
<li><span><a href="_cover_lp__Matrix_rp.html" title="get the covering free module">cover(Matrix)</a> -- get the covering free module</span></li>
<li><span><a href="_cover__Map_lp__Module_rp.html" title="the surjective map from a free module to a module corresponding to the generators">coverMap(Module)</a> -- the surjective map from a free module to a module corresponding to the generators</span></li>
<li><span><a href="_diagonal__Matrix.html" title="make a diagonal matrix">diagonalMatrix</a> -- make a diagonal matrix</span></li>
<li><span><a href="_diff_sq_lp__Matrix_cm__Matrix_rp.html" title="differentiate a matrix by a matrix, the dual notion">diff'(Matrix,Matrix)</a> -- differentiate a matrix by a matrix, the dual notion</span></li>
<li><span><a href="_diff_lp__Matrix_cm__Matrix_rp.html" title="differentiate a matrix by a matrix">diff(Matrix,Matrix)</a> -- differentiate a matrix by a matrix</span></li>
<li><span>divideByVariable(Matrix,RingElement), see <span><a href="_divide__By__Variable.html" title="divide all columns by a (power of a) variable">divideByVariable</a> -- divide all columns by a (power of a) variable</span></span></li>
<li><span>divideByVariable(Matrix,RingElement,ZZ), see <span><a href="_divide__By__Variable.html" title="divide all columns by a (power of a) variable">divideByVariable</a> -- divide all columns by a (power of a) variable</span></span></li>
<li><span><a href="_dual_lp__Matrix_rp.html" title="dual of a map">dual(Matrix)</a> -- dual of a map</span></li>
<li><span><a href="___Ext^__Z__Z_lp__Matrix_cm__Module_rp.html" title="map between Ext modules">Ext^ZZ(Matrix,Module)</a> -- map between Ext modules</span></li>
<li><span><a href="___Ext^__Z__Z_lp__Module_cm__Matrix_rp.html" title="map between Ext modules">Ext^ZZ(Module,Matrix)</a> -- map between Ext modules</span></li>
<li><span><a href="_exterior__Power_lp__Z__Z_cm__Matrix_rp.html" title="exterior power of a matrix">exteriorPower(ZZ,Matrix)</a> -- exterior power of a matrix</span></li>
<li><span><a href="_flatten_lp__Matrix_rp.html" title="puts the columns of a matrix into a single row">flatten(Matrix)</a> -- puts the columns of a matrix into a single row</span></li>
<li><span><a href="_flip_lp__Module_cm__Module_rp.html" title="matrix of commutativity of tensor product">flip(Module,Module)</a> -- matrix of commutativity of tensor product</span></li>
<li><span><a href="_generators_lp__Groebner__Basis_rp.html" title="the generator matrix of a Gröbner basis">generators(GroebnerBasis)</a> -- the generator matrix of a Gröbner basis</span></li>
<li><span><a href="_generators_lp__Ideal_rp.html" title="the generator matrix of an ideal">generators(Ideal)</a> -- the generator matrix of an ideal</span></li>
<li><span><a href="_generators_lp__Module_rp.html" title="the generator matrix of a module">generators(Module)</a> -- the generator matrix of a module</span></li>
<li><span><a href="_generic__Matrix.html" title="make a generic matrix of variables">genericMatrix</a> -- make a generic matrix of variables</span></li>
<li><span><a href="_generic__Skew__Matrix.html" title="make a generic skew symmetric matrix of variables">genericSkewMatrix</a> -- make a generic skew symmetric matrix of variables</span></li>
<li><span><a href="_generic__Symmetric__Matrix.html" title="make a generic symmetric matrix">genericSymmetricMatrix</a> -- make a generic symmetric matrix</span></li>
<li><span>getChangeMatrix(GroebnerBasis), see <span><a href="_get__Change__Matrix.html" title="get the change of basis matrix">getChangeMatrix</a> -- get the change of basis matrix</span></span></li>
<li><span><a href="___H__H^__Z__Z_sp__Chain__Complex__Map.html" title="cohomology of a chain complex map">HH^ZZ ChainComplexMap</a> -- cohomology of a chain complex map</span></li>
<li><span><a href="___H__H_us__Z__Z_sp__Chain__Complex__Map.html" title="homology of a chain complex map">HH_ZZ ChainComplexMap</a> -- homology of a chain complex map</span></li>
<li><span><a href="___Hom_lp__Matrix_cm__Module_rp.html" title="induced map on Hom modules">Hom(Matrix,Module)</a> -- induced map on Hom modules</span></li>
<li><span>Hom(Module,Matrix), see <span><a href="___Hom_lp__Matrix_cm__Module_rp.html" title="induced map on Hom modules">Hom(Matrix,Module)</a> -- induced map on Hom modules</span></span></li>
<li><span>homogenize(Matrix,RingElement), see <span><a href="_homogenize.html" title="homogenize with respect to a variable">homogenize</a> -- homogenize with respect to a variable</span></span></li>
<li><span>homogenize(Matrix,RingElement,List), see <span><a href="_homogenize.html" title="homogenize with respect to a variable">homogenize</a> -- homogenize with respect to a variable</span></span></li>
<li><span>homomorphism(Matrix), see <span><a href="_homomorphism.html" title="get the homomorphism from element of Hom">homomorphism</a> -- get the homomorphism from element of Hom</span></span></li>
<li><span>icFractions(Ring), see <span><a href="../../IntegralClosure/html/_ic__Fractions.html" title="fractions integral over an affine domain">icFractions</a> -- fractions integral over an affine domain</span></span></li>
<li><span><a href="_induced__Map_lp__Module_cm__Module_rp.html" title="compute the map induced by the identity">inducedMap(Module,Module)</a> -- compute the map induced by the identity</span></li>
<li><span><a href="_induced__Map_lp__Module_cm__Module_cm__Matrix_rp.html" title="compute the induced map">inducedMap(Module,Module,Matrix)</a> -- compute the induced map</span></li>
<li><span><a href="_jacobian_lp__Ideal_rp.html" title="the Jacobian matrix of the generators of an ideal">jacobian(Ideal)</a> -- the Jacobian matrix of the generators of an ideal</span></li>
<li><span>jacobian(MonomialIdeal), see <span><a href="_jacobian_lp__Ideal_rp.html" title="the Jacobian matrix of the generators of an ideal">jacobian(Ideal)</a> -- the Jacobian matrix of the generators of an ideal</span></span></li>
<li><span><a href="_jacobian_lp__Matrix_rp.html" title="the matrix of partial derivatives of polynomials in a matrix">jacobian(Matrix)</a> -- the matrix of partial derivatives of polynomials in a matrix</span></li>
<li><span><a href="_jacobian_lp__Ring_rp.html" title="the Jacobian matrix of the polynomials defining a quotient ring">jacobian(Ring)</a> -- the Jacobian matrix of the polynomials defining a quotient ring</span></li>
<li><span><a href="_koszul_lp__Z__Z_cm__Matrix_rp.html" title="a differential in a Koszul complex">koszul(ZZ,Matrix)</a> -- a differential in a Koszul complex</span></li>
<li><span><a href="_lead__Term_lp__Ideal_rp.html" title="get the ideal of greatest terms">leadTerm(Ideal)</a> -- get the ideal of greatest terms</span></li>
<li><span><a href="_lead__Term_lp__Matrix_rp.html" title="get the greatest term of each column">leadTerm(Matrix)</a> -- get the greatest term of each column</span></li>
<li><span><a href="_lead__Term_lp__Z__Z_cm__Ideal_rp.html" title="get the ideal of lead polynomials">leadTerm(ZZ,Ideal)</a> -- get the ideal of lead polynomials</span></li>
<li><span><a href="_lead__Term_lp__Z__Z_cm__Matrix_rp.html" title="get the matrix of lead polynomials of each column">leadTerm(ZZ,Matrix)</a> -- get the matrix of lead polynomials of each column</span></li>
<li><span>lift(Matrix,type of Number), see <span><a href="_lift.html" title="lift to another ring">lift</a> -- lift to another ring</span></span></li>
<li><span>lift(Matrix,type of RingElement), see <span><a href="_lift.html" title="lift to another ring">lift</a> -- lift to another ring</span></span></li>
<li><span><a href="_map_lp__Matrix_rp.html" title="make a matrix with a different degree">map(Matrix)</a> -- make a matrix with a different degree</span></li>
<li><span><a href="_map_lp__Module_rp.html" title="identity map">map(Module)</a> -- identity map</span></li>
<li><span><a href="_map_lp__Module_cm__Module_cm__Function_rp.html" title="create a matrix by specifying a function that gives each entry">map(Module,Module,Function)</a> -- create a matrix by specifying a function that gives each entry</span></li>
<li><span><a href="_map_lp__Module_cm__Module_cm__List_rp.html" title="create a matrix by giving a sparse or dense list of entries">map(Module,Module,List)</a> -- create a matrix by giving a sparse or dense list of entries</span></li>
<li><span><a href="_map_lp__Module_cm__Module_cm__Matrix_rp.html" title="create the matrix induced on generators by a given matrix">map(Module,Module,Matrix)</a> -- create the matrix induced on generators by a given matrix</span></li>
<li><span>map(Module,Module,Number), see <span><a href="_map_lp__Module_cm__Module_cm__Ring__Element_rp.html" title="construct the map induced by multiplication by a ring element on the generators">map(Module,Module,RingElement)</a> -- construct the map induced by multiplication by a ring element on the generators</span></span></li>
<li><span><a href="_map_lp__Module_cm__Module_cm__Ring__Element_rp.html" title="construct the map induced by multiplication by a ring element on the generators">map(Module,Module,RingElement)</a> -- construct the map induced by multiplication by a ring element on the generators</span></li>
<li><span>map(Module,Module,ZZ), see <span><a href="_map_lp__Module_cm__Module_cm__Ring__Element_rp.html" title="construct the map induced by multiplication by a ring element on the generators">map(Module,Module,RingElement)</a> -- construct the map induced by multiplication by a ring element on the generators</span></span></li>
<li><span>map(Module,ZZ,ZZ), see <span><a href="_map_lp__Module_cm__Module_cm__Ring__Element_rp.html" title="construct the map induced by multiplication by a ring element on the generators">map(Module,Module,RingElement)</a> -- construct the map induced by multiplication by a ring element on the generators</span></span></li>
<li><span>map(Module,Module,RingMap,List), see <span><a href="_map_lp__Module_cm__Module_cm__Ring__Map_cm__Matrix_rp.html" title="homomorphism of modules over different rings">map(Module,Module,RingMap,Matrix)</a> -- homomorphism of modules over different rings</span></span></li>
<li><span><a href="_map_lp__Module_cm__Module_cm__Ring__Map_cm__Matrix_rp.html" title="homomorphism of modules over different rings">map(Module,Module,RingMap,Matrix)</a> -- homomorphism of modules over different rings</span></li>
<li><span>map(Module,Nothing,RingMap,List), see <span><a href="_map_lp__Module_cm__Module_cm__Ring__Map_cm__Matrix_rp.html" title="homomorphism of modules over different rings">map(Module,Module,RingMap,Matrix)</a> -- homomorphism of modules over different rings</span></span></li>
<li><span>map(Module,Nothing,RingMap,Matrix), see <span><a href="_map_lp__Module_cm__Module_cm__Ring__Map_cm__Matrix_rp.html" title="homomorphism of modules over different rings">map(Module,Module,RingMap,Matrix)</a> -- homomorphism of modules over different rings</span></span></li>
<li><span>map(Module,RingMap), see <span><a href="_map_lp__Module_cm__Module_cm__Ring__Map_cm__Matrix_rp.html" title="homomorphism of modules over different rings">map(Module,Module,RingMap,Matrix)</a> -- homomorphism of modules over different rings</span></span></li>
<li><span><a href="_map_lp__Module_cm__Nothing_cm__List_rp.html" title="create a matrix by giving a doubly nested list of ring elements">map(Module,Nothing,List)</a> -- create a matrix by giving a doubly nested list of ring elements</span></li>
<li><span><a href="_map_lp__Module_cm__Nothing_cm__Matrix_rp.html" title="recast a matrix to have a new target, and a free module as source">map(Module,Nothing,Matrix)</a> -- recast a matrix to have a new target, and a free module as source</span></li>
<li><span><a href="_map_lp__Module_cm__Z__Z_cm__Function_rp.html" title="create a matrix from a free module by specifying a function that gives each entry">map(Module,ZZ,Function)</a> -- create a matrix from a free module by specifying a function that gives each entry</span></li>
<li><span><a href="_map_lp__Module_cm__Z__Z_cm__List_rp.html" title="create a matrix by giving a sparse or dense list of entries">map(Module,ZZ,List)</a> -- create a matrix by giving a sparse or dense list of entries</span></li>
<li><span><a href="___Matrix_sp_pc_sp__Groebner__Basis.html" title="calculate the normal form of ring elements and matrices using a (partially computed) Gröbner basis">Matrix % GroebnerBasis</a> -- calculate the normal form of ring elements and matrices using a (partially computed) Gröbner basis</span></li>
<li><span><a href="___Matrix_sp_st_sp__Matrix.html" title="matrix multiplication">Matrix * Matrix</a> -- matrix multiplication</span></li>
<li><span><a href="___Matrix_sp_st_st_sp__Matrix.html" title="tensor product">Matrix ** Matrix</a> -- tensor product</span></li>
<li><span><a href="___Matrix_sp_st_st_sp__Module.html" title="tensor product">Matrix ** Module</a> -- tensor product</span></li>
<li><span>Module ** Matrix, see <span><a href="___Matrix_sp_st_st_sp__Module.html" title="tensor product">Matrix ** Module</a> -- tensor product</span></span></li>
<li><span><a href="___Matrix_sp_st_st_sp__Ring.html" title="tensor product">Matrix ** Ring</a> -- tensor product</span></li>
<li><span>Ring ** Matrix, see <span><a href="___Matrix_sp_st_st_sp__Ring.html" title="tensor product">Matrix ** Ring</a> -- tensor product</span></span></li>
<li><span><a href="___Matrix_sp_pl_pl_sp__Matrix.html" title="direct sum of maps">Matrix ++ Matrix</a> -- direct sum of maps</span></li>
<li><span>Matrix ++ RingElement, see <span><a href="___Matrix_sp_pl_pl_sp__Matrix.html" title="direct sum of maps">Matrix ++ Matrix</a> -- direct sum of maps</span></span></li>
<li><span>RingElement ++ Matrix, see <span><a href="___Matrix_sp_pl_pl_sp__Matrix.html" title="direct sum of maps">Matrix ++ Matrix</a> -- direct sum of maps</span></span></li>
<li><span>RingElement ++ RingElement, see <span><a href="___Matrix_sp_pl_pl_sp__Matrix.html" title="direct sum of maps">Matrix ++ Matrix</a> -- direct sum of maps</span></span></li>
<li><span>Matrix // GroebnerBasis, see <span><a href="___Matrix_sp_sl_sl_sp__Matrix.html" title="factor a map through another">Matrix // Matrix</a> -- factor a map through another</span></span></li>
<li><span><a href="___Matrix_sp_sl_sl_sp__Matrix.html" title="factor a map through another">Matrix // Matrix</a> -- factor a map through another</span></li>
<li><span>Matrix // MonomialIdeal, see <span><a href="___Matrix_sp_sl_sl_sp__Matrix.html" title="factor a map through another">Matrix // Matrix</a> -- factor a map through another</span></span></li>
<li><span>RingElement // GroebnerBasis, see <span><a href="___Matrix_sp_sl_sl_sp__Matrix.html" title="factor a map through another">Matrix // Matrix</a> -- factor a map through another</span></span></li>
<li><span><a href="___Matrix_sp^_sp__Array.html" title="component of map corresponding to summand of target">Matrix ^ Array</a> -- component of map corresponding to summand of target</span></li>
<li><span><a href="___Matrix_sp^_sp__List.html" title="select rows">Matrix ^ List</a> -- select rows</span></li>
<li><span><a href="___Matrix_sp^_sp__Z__Z.html" title="power">Matrix ^ ZZ</a> -- power</span></li>
<li><span><a href="___Matrix_sp_us_sp__Array.html" title="component of map corresponding to summand of source">Matrix _ Array</a> -- component of map corresponding to summand of source</span></li>
<li><span><a href="___Matrix_sp_us_sp__List.html" title="select columns">Matrix _ List</a> -- select columns</span></li>
<li><span><a href="___Matrix_sp_vb_sp__Matrix.html" title="join matrices horizontally">Matrix | Matrix</a> -- join matrices horizontally</span></li>
<li><span>RingElement | RingElement, see <span><a href="___Matrix_sp_vb_sp__Matrix.html" title="join matrices horizontally">Matrix | Matrix</a> -- join matrices horizontally</span></span></li>
<li><span><a href="___Matrix_sp_vb_vb_sp__Matrix.html" title="join matrices vertically">Matrix || Matrix</a> -- join matrices vertically</span></li>
<li><span>RingElement || RingElement, see <span><a href="___Matrix_sp_vb_vb_sp__Matrix.html" title="join matrices vertically">Matrix || Matrix</a> -- join matrices vertically</span></span></li>
<li><span><a href="_matrix_lp__List_rp.html" title="create a matrix from a doubly-nested list of ring elements or matrices">matrix(List)</a> -- create a matrix from a doubly-nested list of ring elements or matrices</span></li>
<li><span><a href="_matrix_lp__Matrix_rp.html" title="the matrix between generators">matrix(Matrix)</a> -- the matrix between generators</span></li>
<li><span><a href="_matrix_lp__Ring_cm__List_rp.html" title="create a matrix from a doubly nested list of ring elements or matrices">matrix(Ring,List)</a> -- create a matrix from a doubly nested list of ring elements or matrices</span></li>
<li><span>matrix(RingFamily,List), see <span><a href="_matrix_lp__Ring_cm__List_rp.html" title="create a matrix from a doubly nested list of ring elements or matrices">matrix(Ring,List)</a> -- create a matrix from a doubly nested list of ring elements or matrices</span></span></li>
<li><span><a href="../../Classic/html/_matrix_lp__String_rp.html" title="make a matrix using classic Macaulay syntax">matrix(String)</a> -- make a matrix using classic Macaulay syntax</span></li>
<li><span>Matrix % Ideal, see <span><a href="_methods_spfor_spnormal_spforms_spand_spremainder.html" title="calculate the normal form of ring elements and matrices">methods for normal forms and remainder</a> -- calculate the normal form of ring elements and matrices</span></span></li>
<li><span>Matrix % Matrix, see <span><a href="_methods_spfor_spnormal_spforms_spand_spremainder.html" title="calculate the normal form of ring elements and matrices">methods for normal forms and remainder</a> -- calculate the normal form of ring elements and matrices</span></span></li>
<li><span>Matrix % Module, see <span><a href="_methods_spfor_spnormal_spforms_spand_spremainder.html" title="calculate the normal form of ring elements and matrices">methods for normal forms and remainder</a> -- calculate the normal form of ring elements and matrices</span></span></li>
<li><span>Matrix % MonomialIdeal, see <span><a href="_methods_spfor_spnormal_spforms_spand_spremainder.html" title="calculate the normal form of ring elements and matrices">methods for normal forms and remainder</a> -- calculate the normal form of ring elements and matrices</span></span></li>
<li><span><a href="_mingens_lp__Groebner__Basis_rp.html" title="(partially constructed) minimal generator matrix">mingens(GroebnerBasis)</a> -- (partially constructed) minimal generator matrix</span></li>
<li><span>mingens(Ideal), see <span><a href="_mingens_lp__Module_rp.html" title="minimal generator matrix">mingens(Module)</a> -- minimal generator matrix</span></span></li>
<li><span><a href="_mingens_lp__Module_rp.html" title="minimal generator matrix">mingens(Module)</a> -- minimal generator matrix</span></li>
<li><span><a href="_minimal__Presentation_lp__Matrix_rp.html" title="minimally present source and target of a matrix">minimalPresentation(Matrix)</a> -- minimally present source and target of a matrix</span></li>
<li><span>prune(Matrix), see <span><a href="_minimal__Presentation_lp__Matrix_rp.html" title="minimally present source and target of a matrix">minimalPresentation(Matrix)</a> -- minimally present source and target of a matrix</span></span></li>
<li><span><a href="___Module_sp^_sp__Array.html" title="projection onto summand">Module ^ Array</a> -- projection onto summand</span></li>
<li><span><a href="___Module_sp^_sp__List.html" title="projection onto summand">Module ^ List</a> -- projection onto summand</span></li>
<li><span><a href="___Module_sp_us_sp__Array.html" title="inclusion from summand">Module _ Array</a> -- inclusion from summand</span></li>
<li><span><a href="___Module_sp_us_sp__List.html" title="map from free module to some generators">Module _ List</a> -- map from free module to some generators</span></li>
<li><span><a href="_modulo_lp__Matrix_cm__Matrix_rp.html" title="find the pre-image (pullback) of image of a map (low level version)">modulo(Matrix,Matrix)</a> -- find the pre-image (pullback) of image of a map (low level version)</span></li>
<li><span>modulo(Matrix,Nothing), see <span><a href="_modulo_lp__Matrix_cm__Matrix_rp.html" title="find the pre-image (pullback) of image of a map (low level version)">modulo(Matrix,Matrix)</a> -- find the pre-image (pullback) of image of a map (low level version)</span></span></li>
<li><span>modulo(Nothing,Matrix), see <span><a href="_modulo_lp__Matrix_cm__Matrix_rp.html" title="find the pre-image (pullback) of image of a map (low level version)">modulo(Matrix,Matrix)</a> -- find the pre-image (pullback) of image of a map (low level version)</span></span></li>
<li><span><a href="_presentation_lp__Module_rp.html" title="presentation of a module">presentation(Module)</a> -- presentation of a module</span></li>
<li><span>presentation(QuotientRing), see <span><a href="_presentation_lp__Polynomial__Ring_cm__Quotient__Ring_rp.html" title="presentation of a quotient ring">presentation(PolynomialRing,QuotientRing)</a> -- presentation of a quotient ring</span></span></li>
<li><span>quotient'(Matrix,Matrix), see <span><a href="_quotient_sq.html" title="matrix quotient (opposite)">quotient'</a> -- matrix quotient (opposite)</span></span></li>
<li><span><a href="_quotient_lp__Matrix_cm__Groebner__Basis_rp.html" title="matrix quotient">quotient(Matrix,GroebnerBasis)</a> -- matrix quotient</span></li>
<li><span>quotient(Matrix,Matrix), see <span><a href="_quotient_lp__Matrix_cm__Groebner__Basis_rp.html" title="matrix quotient">quotient(Matrix,GroebnerBasis)</a> -- matrix quotient</span></span></li>
<li><span>quotientRemainder(Matrix,GroebnerBasis), see <span><a href="_quotient__Remainder.html" title="matrix quotient and remainder">quotientRemainder</a> -- matrix quotient and remainder</span></span></li>
<li><span>quotientRemainder(Matrix,Matrix), see <span><a href="_quotient__Remainder.html" title="matrix quotient and remainder">quotientRemainder</a> -- matrix quotient and remainder</span></span></li>
<li><span>quotientRemainder'(Matrix,Matrix), see <span><a href="_quotient__Remainder_sq.html" title="matrix quotient and remainder (opposite)">quotientRemainder'</a> -- matrix quotient and remainder (opposite)</span></span></li>
<li><span><a href="_random_lp__Module_cm__Module_rp.html" title="make a random module map">random(Module,Module)</a> -- make a random module map</span></li>
<li><span>relations(Module), see <span><a href="_relations.html" title="the defining relations">relations</a> -- the defining relations</span></span></li>
<li><span>remainder(Matrix,GroebnerBasis), see <span><a href="_remainder.html" title="matrix remainder">remainder</a> -- matrix remainder</span></span></li>
<li><span>remainder(Matrix,Matrix), see <span><a href="_remainder.html" title="matrix remainder">remainder</a> -- matrix remainder</span></span></li>
<li><span>remainder'(Matrix,Matrix), see <span><a href="_remainder_sq.html" title="matrix quotient and remainder (opposite)">remainder'</a> -- matrix quotient and remainder (opposite)</span></span></li>
<li><span><a href="_reshape_lp__Module_cm__Module_cm__Matrix_rp.html" title="reshape a matrix">reshape(Module,Module,Matrix)</a> -- reshape a matrix</span></li>
<li><span><a href="___Ring__Map_sp_st_st_sp__Matrix.html" title="tensor product of a module map via a ring map">RingMap ** Matrix</a> -- tensor product of a module map via a ring map</span></li>
<li><span>RingMap Matrix, see <span><a href="___Ring__Map_sp__Ring__Element.html" title="apply a ring map">RingMap RingElement</a> -- apply a ring map</span></span></li>
<li><span><a href="_schreyer__Order_lp__Matrix_rp.html" title="create a matrix with the same entries whose source free module has a Schreyer monomial order">schreyerOrder(Matrix)</a> -- create a matrix with the same entries whose source free module has a Schreyer monomial order</span></li>
<li><span><a href="_schreyer__Order_lp__Module_rp.html" title="obtain Schreyer order information">schreyerOrder(Module)</a> -- obtain Schreyer order information</span></li>
<li><span>selectInSubring(ZZ,Matrix), see <span><a href="_select__In__Subring.html" title="select columns in a subring">selectInSubring</a> -- select columns in a subring</span></span></li>
<li><span><a href="_submatrix.html" title="">submatrix</a></span></li>
<li><span><a href="_submatrix_sq.html" title="exclude rows and/or columns of a matrix">submatrix'</a> -- exclude rows and/or columns of a matrix</span></li>
<li><span>substitute(Matrix,List), see <span><a href="_substitute.html" title="substituting values for variables">substitute</a> -- substituting values for variables</span></span></li>
<li><span>substitute(Matrix,Matrix), see <span><a href="_substitute.html" title="substituting values for variables">substitute</a> -- substituting values for variables</span></span></li>
<li><span>substitute(Matrix,Ring), see <span><a href="_substitute.html" title="substituting values for variables">substitute</a> -- substituting values for variables</span></span></li>
<li><span>substitute(Matrix,RingFamily), see <span><a href="_substitute.html" title="substituting values for variables">substitute</a> -- substituting values for variables</span></span></li>
<li><span>substitute(Matrix,ZZ), see <span><a href="_substitute.html" title="substituting values for variables">substitute</a> -- substituting values for variables</span></span></li>
<li><span><a href="_sum_lp__Chain__Complex__Map_rp.html" title="direct sum of the components of a chain map">sum(ChainComplexMap)</a> -- direct sum of the components of a chain map</span></li>
<li><span>super(Matrix), see <span><a href="_super.html" title="get the ambient module">super</a> -- get the ambient module</span></span></li>
<li><span><a href="_symmetric__Power_lp__Z__Z_cm__Matrix_rp.html" title="symmetric power">symmetricPower(ZZ,Matrix)</a> -- symmetric power</span></li>
<li><span><a href="_syz_lp__Groebner__Basis_rp.html" title="retrieve the syzygy matrix">syz(GroebnerBasis)</a> -- retrieve the syzygy matrix</span></li>
<li><span><a href="_syz_lp__Matrix_rp.html" title="compute the syzygy matrix">syz(Matrix)</a> -- compute the syzygy matrix</span></li>
<li><span>tensor(RingMap,Matrix), see <span><a href="_tensor_lp__Ring_cm__Ring__Map_cm__Matrix_rp.html" title="tensor product via a ring map">tensor(Ring,RingMap,Matrix)</a> -- tensor product via a ring map</span></span></li>
<li><span>tensorAssociativity(Module,Module,Module), see <span><a href="_tensor__Associativity.html" title="associativity isomorphisms for tensor products">tensorAssociativity</a> -- associativity isomorphisms for tensor products</span></span></li>
<li><span><a href="_transpose_lp__Matrix_rp.html" title="transpose a matrix">transpose(Matrix)</a> -- transpose a matrix</span></li>
<li><span>universalEmbedding(Ideal), see <span><a href="../../ReesAlgebra/html/_universal__Embedding.html" title="Compute the universal embedding">universalEmbedding</a> -- Compute the universal embedding</span></span></li>
<li><span>universalEmbedding(Module), see <span><a href="../../ReesAlgebra/html/_universal__Embedding.html" title="Compute the universal embedding">universalEmbedding</a> -- Compute the universal embedding</span></span></li>
<li><span><a href="_vars_lp__Ring_rp.html" title="row matrix of the variables">vars(Ring)</a> -- row matrix of the variables</span></li>
<li><span><a href="_wedge__Product_lp__Z__Z_cm__Z__Z_cm__Module_rp.html" title="the exterior multiplication map">wedgeProduct(ZZ,ZZ,Module)</a> -- the exterior multiplication map</span></li>
</ul>
<h2>Methods that use a matrix :</h2>
<ul><li><span>Matrix * Number, see <span><a href="__st.html" title="a binary operator, usually used for multiplication">*</a> -- a binary operator, usually used for multiplication</span></span></li>
<li><span>Matrix * RingElement, see <span><a href="__st.html" title="a binary operator, usually used for multiplication">*</a> -- a binary operator, usually used for multiplication</span></span></li>
<li><span>Matrix * Vector, see <span><a href="__st.html" title="a binary operator, usually used for multiplication">*</a> -- a binary operator, usually used for multiplication</span></span></li>
<li><span>Matrix * ZZ, see <span><a href="__st.html" title="a binary operator, usually used for multiplication">*</a> -- a binary operator, usually used for multiplication</span></span></li>
<li><span>Number * Matrix, see <span><a href="__st.html" title="a binary operator, usually used for multiplication">*</a> -- a binary operator, usually used for multiplication</span></span></li>
<li><span>RingElement * Matrix, see <span><a href="__st.html" title="a binary operator, usually used for multiplication">*</a> -- a binary operator, usually used for multiplication</span></span></li>
<li><span>Matrix + Number, see <span><a href="__pl.html" title="a unary or binary operator, usually used for addition">+</a> -- a unary or binary operator, usually used for addition</span></span></li>
<li><span>Matrix + RingElement, see <span><a href="__pl.html" title="a unary or binary operator, usually used for addition">+</a> -- a unary or binary operator, usually used for addition</span></span></li>
<li><span>Number + Matrix, see <span><a href="__pl.html" title="a unary or binary operator, usually used for addition">+</a> -- a unary or binary operator, usually used for addition</span></span></li>
<li><span>RingElement + Matrix, see <span><a href="__pl.html" title="a unary or binary operator, usually used for addition">+</a> -- a unary or binary operator, usually used for addition</span></span></li>
<li><span>Matrix - Number, see <span><a href="_-.html" title="a unary or binary operator, usually used for negation or subtraction">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li>
<li><span>Matrix - RingElement, see <span><a href="_-.html" title="a unary or binary operator, usually used for negation or subtraction">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li>
<li><span>Number - Matrix, see <span><a href="_-.html" title="a unary or binary operator, usually used for negation or subtraction">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li>
<li><span>RingElement - Matrix, see <span><a href="_-.html" title="a unary or binary operator, usually used for negation or subtraction">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li>
<li><span>Matrix // ZZ, see <span><a href="__sl_sl.html" title="a binary operator, usually used for quotient">//</a> -- a binary operator, usually used for quotient</span></span></li>
<li><span>ZZ // Matrix, see <span><a href="__sl_sl.html" title="a binary operator, usually used for quotient">//</a> -- a binary operator, usually used for quotient</span></span></li>
<li><span>Matrix == Matrix, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>Matrix == Number, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>Matrix == RingElement, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>Matrix == ZZ, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>Number == Matrix, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>RingElement == Matrix, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span><a href="_betti_lp__Matrix_rp.html" title="display of the degrees of a map">betti(Matrix)</a> -- display of the degrees of a map</span></li>
<li><span><a href="_chain__Complex_lp__Matrix_rp.html" title="make a small chain complex">chainComplex(Matrix)</a> -- make a small chain complex</span></li>
<li><span>clean(RR,Matrix), see <span><a href="_clean.html" title="Set to zero elements that are approximately zero">clean</a> -- Set to zero elements that are approximately zero</span></span></li>
<li><span>coefficients(Matrix), see <span><a href="_coefficients.html" title="monomials and their coefficients">coefficients</a> -- monomials and their coefficients</span></span></li>
<li><span>coimage(Matrix), see <span><a href="_coimage.html" title="coimage of a map">coimage</a> -- coimage of a map</span></span></li>
<li><span>cokernel(Matrix), see <span><a href="_cokernel.html" title="cokernel of a map of modules, graded modules, or chaincomplexes">cokernel</a> -- cokernel of a map of modules, graded modules, or chaincomplexes</span></span></li>
<li><span>components(Matrix), see <span><a href="_components.html" title="list the components of a direct sum">components</a> -- list the components of a direct sum</span></span></li>
<li><span>contract(Matrix,Number), see <span><a href="_contract_lp__Matrix_cm__Matrix_rp.html" title="contract a matrix by a matrix">contract(Matrix,Matrix)</a> -- contract a matrix by a matrix</span></span></li>
<li><span>contract(Matrix,RingElement), see <span><a href="_contract_lp__Matrix_cm__Matrix_rp.html" title="contract a matrix by a matrix">contract(Matrix,Matrix)</a> -- contract a matrix by a matrix</span></span></li>
<li><span>contract(Matrix,Vector), see <span><a href="_contract_lp__Matrix_cm__Matrix_rp.html" title="contract a matrix by a matrix">contract(Matrix,Matrix)</a> -- contract a matrix by a matrix</span></span></li>
<li><span>contract(Number,Matrix), see <span><a href="_contract_lp__Matrix_cm__Matrix_rp.html" title="contract a matrix by a matrix">contract(Matrix,Matrix)</a> -- contract a matrix by a matrix</span></span></li>
<li><span>contract(RingElement,Matrix), see <span><a href="_contract_lp__Matrix_cm__Matrix_rp.html" title="contract a matrix by a matrix">contract(Matrix,Matrix)</a> -- contract a matrix by a matrix</span></span></li>
<li><span>contract(Vector,Matrix), see <span><a href="_contract_lp__Matrix_cm__Matrix_rp.html" title="contract a matrix by a matrix">contract(Matrix,Matrix)</a> -- contract a matrix by a matrix</span></span></li>
<li><span><a href="_degree_lp__Matrix_rp.html" title="">degree(Matrix)</a></span></li>
<li><span><a href="_degrees_lp__Matrix_rp.html" title="degrees of target and source">degrees(Matrix)</a> -- degrees of target and source</span></li>
<li><span>determinant(Matrix), see <span><a href="_determinant.html" title="determinant of a matrix">determinant</a> -- determinant of a matrix</span></span></li>
<li><span><a href="_diagonal__Matrix_lp__Matrix_rp.html" title="make a diagonal matrix from entries of a matrix">diagonalMatrix(Matrix)</a> -- make a diagonal matrix from entries of a matrix</span></li>
<li><span>diff(Matrix,Vector), see <span><a href="_diff_lp__Matrix_cm__Matrix_rp.html" title="differentiate a matrix by a matrix">diff(Matrix,Matrix)</a> -- differentiate a matrix by a matrix</span></span></li>
<li><span>diff(Vector,Matrix), see <span><a href="_diff_lp__Matrix_cm__Matrix_rp.html" title="differentiate a matrix by a matrix">diff(Matrix,Matrix)</a> -- differentiate a matrix by a matrix</span></span></li>
<li><span><a href="_diff_lp__Matrix_cm__Ring__Element_rp.html" title="differentiation">diff(Matrix,RingElement)</a> -- differentiation</span></li>
<li><span><a href="_diff_lp__Ring__Element_cm__Matrix_rp.html" title="differentiate each entry of a matrix">diff(RingElement,Matrix)</a> -- differentiate each entry of a matrix</span></li>
<li><span>directSum(Matrix), see <span><a href="_direct__Sum.html" title="direct sum of modules or maps">directSum</a> -- direct sum of modules or maps</span></span></li>
<li><span><a href="_eagon__Northcott_lp__Matrix_rp.html" title="Eagon-Northcott complex of a matrix of linear forms">eagonNorthcott(Matrix)</a> -- Eagon-Northcott complex of a matrix of linear forms</span></li>
<li><span>eigenvalues(Matrix), see <span><a href="_eigenvalues.html" title="find eigenvalues of a matrix">eigenvalues</a> -- find eigenvalues of a matrix</span></span></li>
<li><span>eigenvectors(Matrix), see <span><a href="_eigenvectors.html" title="find eigenvectors of a matrix over RR or CC">eigenvectors</a> -- find eigenvectors of a matrix over RR or CC</span></span></li>
<li><span>entries(Matrix), see <span><a href="_entries.html" title="lists the entries of a matrix">entries</a> -- lists the entries of a matrix</span></span></li>
<li><span>Ext^ZZ(Matrix,Ideal), see <span><a href="___Ext^__Z__Z_lp__Matrix_cm__Module_rp.html" title="map between Ext modules">Ext^ZZ(Matrix,Module)</a> -- map between Ext modules</span></span></li>
<li><span>Ext^ZZ(Matrix,Ring), see <span><a href="___Ext^__Z__Z_lp__Matrix_cm__Module_rp.html" title="map between Ext modules">Ext^ZZ(Matrix,Module)</a> -- map between Ext modules</span></span></li>
<li><span>Ext^ZZ(Ideal,Matrix), see <span><a href="___Ext^__Z__Z_lp__Module_cm__Matrix_rp.html" title="map between Ext modules">Ext^ZZ(Module,Matrix)</a> -- map between Ext modules</span></span></li>
<li><span><a href="_extend_lp__Chain__Complex_cm__Chain__Complex_cm__Matrix_rp.html" title="extend a module map to a chain map, if possible">extend(ChainComplex,ChainComplex,Matrix)</a> -- extend a module map to a chain map, if possible</span></li>
<li><span>forceGB(Matrix), see <span><a href="_force__G__B.html" title="declare that the columns of a matrix are a Gröbner basis">forceGB</a> -- declare that the columns of a matrix are a Gröbner basis</span></span></li>
<li><span><tt>fromDual(Matrix)</tt> (missing documentation<!-- tag: (fromDual,Matrix) -->)</span></li>
<li><span>gb(Matrix), see <span><a href="_gb.html" title="compute a Gröbner basis">gb</a> -- compute a Gröbner basis</span></span></li>
<li><span>gbRemove(Matrix), see <span><a href="_gb__Remove.html" title="remove Gröbner basis">gbRemove</a> -- remove Gröbner basis</span></span></li>
<li><span>gbSnapshot(Matrix), see <span><a href="_gb__Snapshot.html" title="the Gröbner basis matrix as so far computed">gbSnapshot</a> -- the Gröbner basis matrix as so far computed</span></span></li>
<li><span>Matrix _ ZZ, see <span><a href="_generators_spof_spideals_spand_spmodules.html" title="">generators of ideals and modules</a></span></span></li>
<li><span><tt>gramm(Matrix)</tt> (missing documentation<!-- tag: (gramm,Matrix) -->)</span></li>
<li><span><tt>hermite(Matrix)</tt> (missing documentation<!-- tag: (hermite,Matrix) -->)</span></li>
<li><span><a href="_homology_lp__Matrix_cm__Matrix_rp.html" title="homology of a pair of maps">homology(Matrix,Matrix)</a> -- homology of a pair of maps</span></li>
<li><span><a href="_ideal_lp__Matrix_rp.html" title="make an ideal">ideal(Matrix)</a> -- make an ideal</span></li>
<li><span>image(Matrix), see <span><a href="_image.html" title="image of a map">image</a> -- image of a map</span></span></li>
<li><span>indices(Matrix), see <span><a href="_indices_lp__Ring__Element_rp.html" title="indices of variables occurring in a polynomial">indices(RingElement)</a> -- indices of variables occurring in a polynomial</span></span></li>
<li><span>inducedMap(Module,Nothing,Matrix), see <span><a href="_induced__Map_lp__Module_cm__Module_cm__Matrix_rp.html" title="compute the induced map">inducedMap(Module,Module,Matrix)</a> -- compute the induced map</span></span></li>
<li><span>inducedMap(Nothing,Module,Matrix), see <span><a href="_induced__Map_lp__Module_cm__Module_cm__Matrix_rp.html" title="compute the induced map">inducedMap(Module,Module,Matrix)</a> -- compute the induced map</span></span></li>
<li><span>inducedMap(Nothing,Nothing,Matrix), see <span><a href="_induced__Map_lp__Module_cm__Module_cm__Matrix_rp.html" title="compute the induced map">inducedMap(Module,Module,Matrix)</a> -- compute the induced map</span></span></li>
<li><span>inducesWellDefinedMap(Module,Module,Matrix), see <span><a href="_induces__Well__Defined__Map.html" title="whether a map is well defined">inducesWellDefinedMap</a> -- whether a map is well defined</span></span></li>
<li><span>inducesWellDefinedMap(Module,Nothing,Matrix), see <span><a href="_induces__Well__Defined__Map.html" title="whether a map is well defined">inducesWellDefinedMap</a> -- whether a map is well defined</span></span></li>
<li><span>inducesWellDefinedMap(Nothing,Module,Matrix), see <span><a href="_induces__Well__Defined__Map.html" title="whether a map is well defined">inducesWellDefinedMap</a> -- whether a map is well defined</span></span></li>
<li><span>inducesWellDefinedMap(Nothing,Nothing,Matrix), see <span><a href="_induces__Well__Defined__Map.html" title="whether a map is well defined">inducesWellDefinedMap</a> -- whether a map is well defined</span></span></li>
<li><span>installHilbertFunction(Matrix,RingElement), see <span><a href="_install__Hilbert__Function.html" title="install a Hilbert function without computation">installHilbertFunction</a> -- install a Hilbert function without computation</span></span></li>
<li><span><a href="_inverse_lp__Matrix_rp.html" title="compute the inverse">inverse(Matrix)</a> -- compute the inverse</span></li>
<li><span>isHomogeneous(Matrix), see <span><a href="_is__Homogeneous.html" title="whether something is homogeneous (graded)">isHomogeneous</a> -- whether something is homogeneous (graded)</span></span></li>
<li><span>isInjective(Matrix), see <span><a href="_is__Injective.html" title="whether a map is injective">isInjective</a> -- whether a map is injective</span></span></li>
<li><span>isIsomorphism(Matrix), see <span><a href="_is__Isomorphism.html" title="whether a map is an isomorphism">isIsomorphism</a> -- whether a map is an isomorphism</span></span></li>
<li><span>isLLL(Matrix), see <span><a href="../../LLLBases/html/_is__L__L__L.html" title="is a basis an LLL basis?">isLLL</a> -- is a basis an LLL basis?</span></span></li>
<li><span>isSurjective(Matrix), see <span><a href="_is__Surjective.html" title="whether a map is surjective">isSurjective</a> -- whether a map is surjective</span></span></li>
<li><span>isWellDefined(Matrix), see <span><a href="_is__Well__Defined.html" title="whether a map is well defined">isWellDefined</a> -- whether a map is well defined</span></span></li>
<li><span><a href="_kernel_lp__Matrix_rp.html" title="kernel of a matrix">kernel(Matrix)</a> -- kernel of a matrix</span></li>
<li><span><tt>kernelLLL(Matrix)</tt> (missing documentation<!-- tag: (kernelLLL,Matrix) -->)</span></li>
<li><span><a href="_koszul_lp__Matrix_rp.html" title="the Koszul complex">koszul(Matrix)</a> -- the Koszul complex</span></li>
<li><span>leadComponent(Matrix), see <span><a href="_lead__Component.html" title="the leading component of a vector or matrix">leadComponent</a> -- the leading component of a vector or matrix</span></span></li>
<li><span>lift(Matrix,type of QQ,type of QQ), see <span><a href="_lift.html" title="lift to another ring">lift</a> -- lift to another ring</span></span></li>
<li><span>lift(Matrix,type of QQ,type of ZZ), see <span><a href="_lift.html" title="lift to another ring">lift</a> -- lift to another ring</span></span></li>
<li><span>lift(Matrix,type of ZZ,type of ZZ), see <span><a href="_lift.html" title="lift to another ring">lift</a> -- lift to another ring</span></span></li>
<li><span>LLL(Matrix), see <span><a href="../../LLLBases/html/___L__L__L.html" title="compute an LLL basis">LLL</a> -- compute an LLL basis</span></span></li>
<li><span>LUdecomposition(Matrix), see <span><a href="___L__Udecomposition.html" title="LU decomposition">LUdecomposition</a> -- LU decomposition</span></span></li>
<li><span><a href="_map_lp__Ring_cm__Matrix_rp.html" title="make a ring map">map(Ring,Matrix)</a> -- make a ring map</span></li>
<li><span><a href="_map_lp__Ring_cm__Ring_cm__Matrix_rp.html" title="make a ring map">map(Ring,Ring,Matrix)</a> -- make a ring map</span></li>
<li><span><a href="_marked__G__B_lp__Matrix_cm__Matrix_rp.html" title="make a marked Gröbner basis">markedGB(Matrix,Matrix)</a> -- make a marked Gröbner basis</span></li>
<li><span><a href="___Matrix_sp_st_st_sp__Ring__Element.html" title="a binary operator, usually used for tensor product or Cartesian product">Matrix ** RingElement</a> -- a binary operator, usually used for tensor product or Cartesian product</span></li>
<li><span>Matrix // RingElement, see <span><a href="___Matrix_sp_sl_sl_sp__Matrix.html" title="factor a map through another">Matrix // Matrix</a> -- factor a map through another</span></span></li>
<li><span>RingElement // Matrix, see <span><a href="___Matrix_sp_sl_sl_sp__Matrix.html" title="factor a map through another">Matrix // Matrix</a> -- factor a map through another</span></span></li>
<li><span><a href="___Matrix_sp_us_sp__Sequence.html" title="get entry of matrix">Matrix _ Sequence</a> -- get entry of matrix</span></li>
<li><span>Matrix | RingElement, see <span><a href="___Matrix_sp_vb_sp__Matrix.html" title="join matrices horizontally">Matrix | Matrix</a> -- join matrices horizontally</span></span></li>
<li><span>Matrix | ZZ, see <span><a href="___Matrix_sp_vb_sp__Matrix.html" title="join matrices horizontally">Matrix | Matrix</a> -- join matrices horizontally</span></span></li>
<li><span>RingElement | Matrix, see <span><a href="___Matrix_sp_vb_sp__Matrix.html" title="join matrices horizontally">Matrix | Matrix</a> -- join matrices horizontally</span></span></li>
<li><span>ZZ | Matrix, see <span><a href="___Matrix_sp_vb_sp__Matrix.html" title="join matrices horizontally">Matrix | Matrix</a> -- join matrices horizontally</span></span></li>
<li><span>Matrix || RingElement, see <span><a href="___Matrix_sp_vb_vb_sp__Matrix.html" title="join matrices vertically">Matrix || Matrix</a> -- join matrices vertically</span></span></li>
<li><span>Matrix || ZZ, see <span><a href="___Matrix_sp_vb_vb_sp__Matrix.html" title="join matrices vertically">Matrix || Matrix</a> -- join matrices vertically</span></span></li>
<li><span>RingElement || Matrix, see <span><a href="___Matrix_sp_vb_vb_sp__Matrix.html" title="join matrices vertically">Matrix || Matrix</a> -- join matrices vertically</span></span></li>
<li><span>ZZ || Matrix, see <span><a href="___Matrix_sp_vb_vb_sp__Matrix.html" title="join matrices vertically">Matrix || Matrix</a> -- join matrices vertically</span></span></li>
<li><span>Matrix % RingElement, see <span><a href="_methods_spfor_spnormal_spforms_spand_spremainder.html" title="calculate the normal form of ring elements and matrices">methods for normal forms and remainder</a> -- calculate the normal form of ring elements and matrices</span></span></li>
<li><span>RingElement % Matrix, see <span><a href="_methods_spfor_spnormal_spforms_spand_spremainder.html" title="calculate the normal form of ring elements and matrices">methods for normal forms and remainder</a> -- calculate the normal form of ring elements and matrices</span></span></li>
<li><span><a href="_minors_lp__Z__Z_cm__Matrix_rp.html" title="ideal generated by minors">minors(ZZ,Matrix)</a> -- ideal generated by minors</span></li>
<li><span><a href="_monomial__Ideal_lp__Matrix_rp.html" title="monomial ideal of lead monomials">monomialIdeal(Matrix)</a> -- monomial ideal of lead monomials</span></li>
<li><span>monomials(Matrix), see <span><a href="_monomials.html" title="matrix of monomials in a ring element or matrix">monomials</a> -- matrix of monomials in a ring element or matrix</span></span></li>
<li><span>mutableMatrix(Matrix), see <span><a href="_mutable__Matrix.html" title="make a mutable matrix">mutableMatrix</a> -- make a mutable matrix</span></span></li>
<li><span>newCoordinateSystem(PolynomialRing,Matrix), see <span><a href="_new__Coordinate__System.html" title="change variables">newCoordinateSystem</a> -- change variables</span></span></li>
<li><span>norm(InfiniteNumber,Matrix), see <span><a href="_norm.html" title="">norm</a></span></span></li>
<li><span>norm(Matrix), see <span><a href="_norm.html" title="">norm</a></span></span></li>
<li><span>norm(RR,Matrix), see <span><a href="_norm.html" title="">norm</a></span></span></li>
<li><span><a href="_num__Columns_lp__Matrix_rp.html" title="number of columns in a matrix or mutable matrix">numColumns(Matrix)</a> -- number of columns in a matrix or mutable matrix</span></li>
<li><span>numeric(Matrix), see <span><a href="_numeric.html" title="convert to floating point">numeric</a> -- convert to floating point</span></span></li>
<li><span>numeric(ZZ,Matrix), see <span><a href="_numeric.html" title="convert to floating point">numeric</a> -- convert to floating point</span></span></li>
<li><span><a href="_num__Rows_lp__Matrix_rp.html" title="number of rows in a matrix or mutable matrix">numRows(Matrix)</a> -- number of rows in a matrix or mutable matrix</span></li>
<li><span>permanents(ZZ,Matrix), see <span><a href="_permanents.html" title="ideal generated by square permanents of a matrix">permanents</a> -- ideal generated by square permanents of a matrix</span></span></li>
<li><span>pfaffians(ZZ,Matrix), see <span><a href="_pfaffians.html" title="ideal generated by Pfaffians">pfaffians</a> -- ideal generated by Pfaffians</span></span></li>
<li><span><a href="_pivots_lp__Matrix_rp.html" title="list of pivot locations of a matrix">pivots(Matrix)</a> -- list of pivot locations of a matrix</span></li>
<li><span>precision(Matrix), see <span><a href="_precision.html" title="">precision</a></span></span></li>
<li><span>rank(Matrix), see <span><a href="_rank.html" title="compute the rank">rank</a> -- compute the rank</span></span></li>
<li><span><a href="_resolution_lp__Matrix_rp.html" title="given a module map represented by a matrix, produce a comparison map between resolutions of its source and target">resolution(Matrix)</a> -- given a module map represented by a matrix, produce a comparison map between resolutions of its source and target</span></li>
<li><span>ring(Matrix), see <span><a href="_ring.html" title="get the associated ring of an object">ring</a> -- get the associated ring of an object</span></span></li>
<li><span>ringFromFractions(Matrix,RingElement), see <span><a href="../../IntegralClosure/html/_ring__From__Fractions.html" title="find presentation for f.g. ring">ringFromFractions</a> -- find presentation for f.g. ring</span></span></li>
<li><span>rsort(Matrix), see <span><a href="_rsort.html" title="sort a list or matrix in reverse order">rsort</a> -- sort a list or matrix in reverse order</span></span></li>
<li><span><a href="../../SimpleDoc/html/_simple__Doc__Frob_lp__Z__Z_cm__Matrix_rp.html" title="A sample documentation node">simpleDocFrob(ZZ,Matrix)</a> -- A sample documentation node</span></li>
<li><span><a href="_smith__Normal__Form_lp__Matrix_rp.html" title="smith normal form for a matrix over ZZ or a PID">smithNormalForm(Matrix)</a> -- smith normal form for a matrix over ZZ or a PID</span></li>
<li><span>solve(Matrix,Matrix), see <span><a href="_solve.html" title="solve a linear equation">solve</a> -- solve a linear equation</span></span></li>
<li><span><a href="_sort_lp__Matrix_rp.html" title="sort the columns of a matrix">sort(Matrix)</a> -- sort the columns of a matrix</span></li>
<li><span>sortColumns(Matrix), see <span><a href="_sort__Columns.html" title="permutation giving sort order">sortColumns</a> -- permutation giving sort order</span></span></li>
<li><span><a href="_source_lp__Matrix_rp.html" title="find the source module of matrix">source(Matrix)</a> -- find the source module of matrix</span></li>
<li><span>submatrix'(Matrix,Nothing,VisibleList), see <span><a href="_submatrix_sq.html" title="exclude rows and/or columns of a matrix">submatrix'</a> -- exclude rows and/or columns of a matrix</span></span></li>
<li><span>submatrix'(Matrix,VisibleList), see <span><a href="_submatrix_sq.html" title="exclude rows and/or columns of a matrix">submatrix'</a> -- exclude rows and/or columns of a matrix</span></span></li>
<li><span>submatrix'(Matrix,VisibleList,Nothing), see <span><a href="_submatrix_sq.html" title="exclude rows and/or columns of a matrix">submatrix'</a> -- exclude rows and/or columns of a matrix</span></span></li>
<li><span>submatrix'(Matrix,VisibleList,VisibleList), see <span><a href="_submatrix_sq.html" title="exclude rows and/or columns of a matrix">submatrix'</a> -- exclude rows and/or columns of a matrix</span></span></li>
<li><span><a href="_submatrix_lp__Matrix_cm__Visible__List_rp.html" title="select columns">submatrix(Matrix,VisibleList)</a> -- select columns</span></li>
<li><span>submatrix(Matrix,Nothing,VisibleList), see <span><a href="_submatrix_lp__Matrix_cm__Visible__List_cm__Visible__List_rp.html" title="select part of a matrix">submatrix(Matrix,VisibleList,VisibleList)</a> -- select part of a matrix</span></span></li>
<li><span>submatrix(Matrix,VisibleList,Nothing), see <span><a href="_submatrix_lp__Matrix_cm__Visible__List_cm__Visible__List_rp.html" title="select part of a matrix">submatrix(Matrix,VisibleList,VisibleList)</a> -- select part of a matrix</span></span></li>
<li><span><a href="_submatrix_lp__Matrix_cm__Visible__List_cm__Visible__List_rp.html" title="select part of a matrix">submatrix(Matrix,VisibleList,VisibleList)</a> -- select part of a matrix</span></li>
<li><span>subquotient(Matrix,Matrix), see <span><a href="_subquotient.html" title="make a subquotient module">subquotient</a> -- make a subquotient module</span></span></li>
<li><span>subquotient(Matrix,Nothing), see <span><a href="_subquotient.html" title="make a subquotient module">subquotient</a> -- make a subquotient module</span></span></li>
<li><span>subquotient(Module,Matrix,Matrix), see <span><a href="_subquotient.html" title="make a subquotient module">subquotient</a> -- make a subquotient module</span></span></li>
<li><span>subquotient(Module,Matrix,Nothing), see <span><a href="_subquotient.html" title="make a subquotient module">subquotient</a> -- make a subquotient module</span></span></li>
<li><span>subquotient(Module,Nothing,Matrix), see <span><a href="_subquotient.html" title="make a subquotient module">subquotient</a> -- make a subquotient module</span></span></li>
<li><span>subquotient(Nothing,Matrix), see <span><a href="_subquotient.html" title="make a subquotient module">subquotient</a> -- make a subquotient module</span></span></li>
<li><span>substitute(Ideal,Matrix), see <span><a href="_substitute.html" title="substituting values for variables">substitute</a> -- substituting values for variables</span></span></li>
<li><span>substitute(Matrix,Option), see <span><a href="_substitute.html" title="substituting values for variables">substitute</a> -- substituting values for variables</span></span></li>
<li><span>substitute(Module,Matrix), see <span><a href="_substitute.html" title="substituting values for variables">substitute</a> -- substituting values for variables</span></span></li>
<li><span>substitute(RingElement,Matrix), see <span><a href="_substitute.html" title="substituting values for variables">substitute</a> -- substituting values for variables</span></span></li>
<li><span>substitute(Vector,Matrix), see <span><a href="_substitute.html" title="substituting values for variables">substitute</a> -- substituting values for variables</span></span></li>
<li><span>support(Matrix), see <span><a href="_support.html" title="list of variables occurring in a polynomial or matrix">support</a> -- list of variables occurring in a polynomial or matrix</span></span></li>
<li><span>SVD(Matrix), see <span><a href="___S__V__D.html" title="singular value decomposition of a matrix">SVD</a> -- singular value decomposition of a matrix</span></span></li>
<li><span>symmetricAlgebra(Matrix), see <span><a href="_symmetric__Algebra.html" title="the symmetric algebra of a module">symmetricAlgebra</a> -- the symmetric algebra of a module</span></span></li>
<li><span>symmetricAlgebra(Nothing,Nothing,Matrix), see <span><a href="_symmetric__Algebra.html" title="the symmetric algebra of a module">symmetricAlgebra</a> -- the symmetric algebra of a module</span></span></li>
<li><span>symmetricAlgebra(Nothing,Ring,Matrix), see <span><a href="_symmetric__Algebra.html" title="the symmetric algebra of a module">symmetricAlgebra</a> -- the symmetric algebra of a module</span></span></li>
<li><span>symmetricAlgebra(Ring,Nothing,Matrix), see <span><a href="_symmetric__Algebra.html" title="the symmetric algebra of a module">symmetricAlgebra</a> -- the symmetric algebra of a module</span></span></li>
<li><span>symmetricAlgebra(Ring,Ring,Matrix), see <span><a href="_symmetric__Algebra.html" title="the symmetric algebra of a module">symmetricAlgebra</a> -- the symmetric algebra of a module</span></span></li>
<li><span>symmetricKernel(Matrix), see <span><a href="../../ReesAlgebra/html/_symmetric__Kernel.html" title="Compute the Rees ring of the image of a matrix">symmetricKernel</a> -- Compute the Rees ring of the image of a matrix</span></span></li>
<li><span><a href="_target_lp__Matrix_rp.html" title="find the target module of matrix">target(Matrix)</a> -- find the target module of matrix</span></li>
<li><span><a href="_tensor_lp__Ring_cm__Ring__Map_cm__Matrix_rp.html" title="tensor product via a ring map">tensor(Ring,RingMap,Matrix)</a> -- tensor product via a ring map</span></li>
<li><span><tt>toDual(ZZ,Matrix)</tt> (missing documentation<!-- tag: (toDual,ZZ,Matrix) -->)</span></li>
<li><span>topCoefficients(Matrix), see <span><a href="_top__Coefficients.html" title="first variable and its coefficient of a polynomial or matrix">topCoefficients</a> -- first variable and its coefficient of a polynomial or matrix</span></span></li>
<li><span><a href="_trace_lp__Matrix_rp.html" title="trace of a matrix">trace(Matrix)</a> -- trace of a matrix</span></li>
</ul>
</div>
<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="___Matrix.html" title="the class of all matrices">Matrix</a> is <span>a <a href="___Type.html">type</a></span>, with ancestor classes <a href="___Module__Map.html" title="the class of all maps between modules">ModuleMap</a> < <a href="___Hash__Table.html" title="the class of all hash tables">HashTable</a> < <a href="___Thing.html" title="the class of all things">Thing</a>.</p>
</div>
</div>
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