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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_matrices.html" title="">matrices</a> > <a href="_basic_sparithmetic_spof_spmatrices.html" title="">basic arithmetic of matrices</a></div>
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<div><h1>basic arithmetic of matrices</h1>
<div><h2>+</h2>
To add two matrices, use the <a href="__pl.html" title="a unary or binary operator, usually used for addition">+</a> operator.<table class="examples"><tr><td><pre>i1 : ff = matrix{{1,2,3},{4,5,6}}
o1 = | 1 2 3 |
| 4 5 6 |
2 3
o1 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i2 : gg = matrix{{4,5,6},{1,2,3}}
o2 = | 4 5 6 |
| 1 2 3 |
2 3
o2 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i3 : ff+gg
o3 = | 5 7 9 |
| 5 7 9 |
2 3
o3 : Matrix ZZ <--- ZZ</pre>
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The matrices in question must have the same number of rows and columns and also must have the same ring.<h2>-</h2>
To subtract two matrices, use the <a href="_-.html" title="a unary or binary operator, usually used for negation or subtraction">-</a> operator.<table class="examples"><tr><td><pre>i4 : ff-gg
o4 = | -3 -3 -3 |
| 3 3 3 |
2 3
o4 : Matrix ZZ <--- ZZ</pre>
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The matrices in question must have the same number of rows and columns and also must have the same ring.<h2>*</h2>
To multiply two matrices use the <a href="__st.html" title="a binary operator, usually used for multiplication">*</a> operator.<table class="examples"><tr><td><pre>i5 : R = ZZ/17[a..l];</pre>
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<tr><td><pre>i6 : ff = matrix {{a,b,c},{d,e,f}}
o6 = | a b c |
| d e f |
2 3
o6 : Matrix R <--- R</pre>
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<tr><td><pre>i7 : gg = matrix {{g,h},{i,j},{k,l}}
o7 = | g h |
| i j |
| k l |
3 2
o7 : Matrix R <--- R</pre>
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<tr><td><pre>i8 : ff * gg
o8 = | ag+bi+ck ah+bj+cl |
| dg+ei+fk dh+ej+fl |
2 2
o8 : Matrix R <--- R</pre>
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<h2>^</h2>
To raise a square matrix to a power, use the <a href="_^.html" title="a binary operator, usually used for powers">^</a> operator.<table class="examples"><tr><td><pre>i9 : ff = matrix{{1,2,3},{4,5,6},{7,8,9}}
o9 = | 1 2 3 |
| 4 5 6 |
| 7 8 9 |
3 3
o9 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i10 : ff^4
o10 = | 7560 9288 11016 |
| 17118 21033 24948 |
| 26676 32778 38880 |
3 3
o10 : Matrix ZZ <--- ZZ</pre>
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<h2>inverse of a matrix</h2>
If a matrix <tt>f</tt> is invertible, then <tt>f^-1</tt> will work.<table class="examples"/>
<h2>==</h2>
To check whether two matrices are equal, one can use <a href="__eq_eq.html" title="equality">==</a>.<table class="examples"><tr><td><pre>i11 : ff == gg
o11 = false</pre>
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<tr><td><pre>i12 : ff == ff
o12 = true</pre>
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However, given two matrices <tt>ff</tt> and <tt>gg</tt>, it can be the case that <tt>ff - gg == 0</tt> returns <a href="_true.html" title="">true</a> but <tt>ff == gg</tt> returns <a href="_false.html" title="">false</a>.<table class="examples"><tr><td><pre>i13 : M = R^{1,2,3}
3
o13 = R
o13 : R-module, free, degrees {-1, -2, -3}</pre>
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<tr><td><pre>i14 : N = R^3
3
o14 = R
o14 : R-module, free</pre>
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<tr><td><pre>i15 : ff = id_M
o15 = {-1} | 1 0 0 |
{-2} | 0 1 0 |
{-3} | 0 0 1 |
3 3
o15 : Matrix R <--- R</pre>
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<tr><td><pre>i16 : gg = id_N
o16 = | 1 0 0 |
| 0 1 0 |
| 0 0 1 |
3 3
o16 : Matrix R <--- R</pre>
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<tr><td><pre>i17 : ff - gg == 0
o17 = true</pre>
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<tr><td><pre>i18 : ff == gg
o18 = false</pre>
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Since the degrees attached to the matrices were different, <a href="__eq_eq.html" title="equality">==</a> returned the value <a href="_false.html" title="">false</a>.<h2>!=</h2>
To check whether two matrices are not equal, one can use <a href="_!_eq.html" title="inequality">!=</a>:<table class="examples"><tr><td><pre>i19 : ff != gg
o19 = true</pre>
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From the definition above of <tt>ff</tt> and <tt>gg</tt> we see that <a href="_!_eq.html" title="inequality">!=</a> will return a value of <a href="_true.html" title="">true</a> if the degrees attached the the matrices are different, even if the entries are the same.<h2>**</h2>
Since tensor product (also known as Kronecker product and outer product) is a functor of two variables, we may compute the tensor product of two matrices. Recalling that a matrix is a map between modules, we may write:<pre>
ff : K ---> L
gg : M ---> N
ff ** gg : K ** M ---> L ** N
</pre>
<table class="examples"><tr><td><pre>i20 : ff = matrix {{a,b,c},{d,e,f}}
o20 = | a b c |
| d e f |
2 3
o20 : Matrix R <--- R</pre>
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<tr><td><pre>i21 : gg = matrix {{g,h},{i,j},{k,l}}
o21 = | g h |
| i j |
| k l |
3 2
o21 : Matrix R <--- R</pre>
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<tr><td><pre>i22 : ff ** gg
o22 = | ag ah bg bh cg ch |
| ai aj bi bj ci cj |
| ak al bk bl ck cl |
| dg dh eg eh fg fh |
| di dj ei ej fi fj |
| dk dl ek el fk fl |
6 6
o22 : Matrix R <--- R</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_extracting_spinformation_spabout_spa_spmatrix.html" title="">extracting information about a matrix</a></span></li>
</ul>
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