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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_basic_spcommutative_spalgebra.html" title="">basic commutative algebra</a> > <a href="___M2__Singular__Book.html" title="Macaulay2 examples for the Singular book">M2SingularBook</a> > <a href="___Singular_sp__Book_sp2.1.6.html" title="matrix operations">Singular Book 2.1.6</a></div>
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<div><h1>Singular Book 2.1.6 -- matrix operations</h1>
<div>In Macaulay2, matrices are defined over a ring. There are many ways to make a matrix, but the easiest is to use the <a href="_matrix.html" title="make a matrix">matrix</a> routine.<table class="examples"><tr><td><pre>i1 : A = QQ[x,y,z];</pre>
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<tr><td><pre>i2 : M = matrix{{1, x+y, z^2},
{x, 0, x*y*z}}
o2 = | 1 x+y z2 |
| x 0 xyz |
2 3
o2 : Matrix A <--- A</pre>
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<tr><td><pre>i3 : N = matrix(A, {{1,2,3},{4,5,6},{7,8,9}})
o3 = | 1 2 3 |
| 4 5 6 |
| 7 8 9 |
3 3
o3 : Matrix A <--- A</pre>
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The usual matrix arithmetic operations work.<table class="examples"><tr><td><pre>i4 : M+M
o4 = | 2 2x+2y 2z2 |
| 2x 0 2xyz |
2 3
o4 : Matrix A <--- A</pre>
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<tr><td><pre>i5 : x*N
o5 = | x 2x 3x |
| 4x 5x 6x |
| 7x 8x 9x |
3 3
o5 : Matrix A <--- A</pre>
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<tr><td><pre>i6 : M*N
o6 = | 7z2+4x+4y+1 8z2+5x+5y+2 9z2+6x+6y+3 |
| 7xyz+x 8xyz+2x 9xyz+3x |
2 3
o6 : Matrix A <--- A</pre>
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<tr><td><pre>i7 : N^3
o7 = | 468 576 684 |
| 1062 1305 1548 |
| 1656 2034 2412 |
3 3
o7 : Matrix A <--- A</pre>
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<tr><td><pre>i8 : ((x+y+z)*N)^3
o8 = | 468x3+1404x2y+1404xy2+
| 1062x3+3186x2y+3186xy2
| 1656x3+4968x2y+4968xy2
------------------------------------------------------------------------
468y3+1404x2z+2808xyz+1404y2z+1404xz2+1404yz2+468z3
+1062y3+3186x2z+6372xyz+3186y2z+3186xz2+3186yz2+1062z3
+1656y3+4968x2z+9936xyz+4968y2z+4968xz2+4968yz2+1656z3
------------------------------------------------------------------------
576x3+1728x2y+1728xy2+576y3+1728x2z+3456xyz+1728y2z+1728xz2+1728yz2+576z
1305x3+3915x2y+3915xy2+1305y3+3915x2z+7830xyz+3915y2z+3915xz2+3915yz2+13
2034x3+6102x2y+6102xy2+2034y3+6102x2z+12204xyz+6102y2z+6102xz2+6102yz2+2
------------------------------------------------------------------------
3 684x3+2052x2y+2052xy2+
05z3 1548x3+4644x2y+4644xy2
034z3 2412x3+7236x2y+7236xy2
------------------------------------------------------------------------
684y3+2052x2z+4104xyz+2052y2z+2052xz2+2052yz2+684z3 |
+1548y3+4644x2z+9288xyz+4644y2z+4644xz2+4644yz2+1548z3 |
+2412y3+7236x2z+14472xyz+7236y2z+7236xz2+7236yz2+2412z3 |
3 3
o8 : Matrix A <--- A</pre>
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Indices in Macaulay2 are always 0 based, so the upper left entry is (0,0). Indexing is performed using <a href="__us.html" title="a binary operator, used for subscripting and access to elements">_</a>.<table class="examples"><tr><td><pre>i9 : M_(1,2)
o9 = x*y*z
o9 : A</pre>
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Matrices cannot be modified. Make a MutableMatrix if you want to modify a matrix.<table class="examples"><tr><td><pre>i10 : M1 = mutableMatrix M
o10 = | 1 x+y z2 |
| x . xyz |
o10 : MutableMatrix</pre>
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<tr><td><pre>i11 : M1_(1,2) = 37_A
o11 = 37
o11 : A</pre>
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<tr><td><pre>i12 : M1
o12 = | 1 x+y z2 |
| x . 37 |
o12 : MutableMatrix</pre>
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<tr><td><pre>i13 : matrix M1
o13 = | 1 x+y z2 |
| x 0 37 |
2 3
o13 : Matrix A <--- A</pre>
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Matrices can be concatenated, either horizontally or vertically. The number of rows must match for horizontal concatenation, and the number of columns must match for vertical concatenation.<table class="examples"><tr><td><pre>i14 : M | M
o14 = | 1 x+y z2 1 x+y z2 |
| x 0 xyz x 0 xyz |
2 6
o14 : Matrix A <--- A</pre>
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<tr><td><pre>i15 : M || N
o15 = | 1 x+y z2 |
| x 0 xyz |
| 1 2 3 |
| 4 5 6 |
| 7 8 9 |
5 3
o15 : Matrix A <--- A</pre>
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Use <a href="_ideal_lp__Matrix_rp.html" title="make an ideal">ideal(Matrix)</a> to obtain the ideal generated by the entries of a matrix.<table class="examples"><tr><td><pre>i16 : ideal M
2
o16 = ideal (1, x, x + y, 0, z , x*y*z)
o16 : Ideal of A</pre>
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The <i>n</i> by <i>n</i> identity matrix is the identity map of the freemodule <i>A<sup>n</sup></i>.<table class="examples"><tr><td><pre>i17 : F = A^5
5
o17 = A
o17 : A-module, free</pre>
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<tr><td><pre>i18 : id_(A^5)
o18 = | 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 1 0 0 |
| 0 0 0 1 0 |
| 0 0 0 0 1 |
5 5
o18 : Matrix A <--- A</pre>
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In Macaulay2, integer matrices are just matrices defined over the ring of integers ZZ.<table class="examples"><tr><td><pre>i19 : matrix{{1,2,3},{4,5,6}}
o19 = | 1 2 3 |
| 4 5 6 |
2 3
o19 : Matrix ZZ <--- ZZ</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_matrices.html" title="">matrices</a></span></li>
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