(Disambiguation: for division of matrices, which can also be thought of as solving a system of linear equations, see instead Matrix // Matrix. For lifting a map between modules to a map between their free resolutions, see extend.)
There are several restrictions. The first is that there are only a limited number of rings for which this function is implemented. Second, over
RR or
CC, the matrix
A must be a square non-singular matrix. Third, if
A and
b are mutable matrices over
RR or
CC, they must be dense matrices.
i1 : kk = ZZ/101;
|
i2 : A = matrix"1,2,3,4;1,3,6,10;19,7,11,13" ** kk
o2 = | 1 2 3 4 |
| 1 3 6 10 |
| 19 7 11 13 |
3 4
o2 : Matrix kk <--- kk
|
i3 : b = matrix"1;1;1" ** kk
o3 = | 1 |
| 1 |
| 1 |
3 1
o3 : Matrix kk <--- kk
|
i4 : x = solve(A,b)
o4 = | 2 |
| -1 |
| 34 |
| 0 |
4 1
o4 : Matrix kk <--- kk
|
i5 : A*x-b
o5 = 0
3 1
o5 : Matrix kk <--- kk
|
Over
RR or
CC, the matrix
A must be a non-singular square matrix.
i6 : printingPrecision = 2;
|
i7 : A = matrix "1,2,3;1,3,6;19,7,11" ** RR
o7 = | 1 2 3 |
| 1 3 6 |
| 19 7 11 |
3 3
o7 : Matrix RR <--- RR
53 53
|
i8 : b = matrix "1;1;1" ** RR
o8 = | 1 |
| 1 |
| 1 |
3 1
o8 : Matrix RR <--- RR
53 53
|
i9 : x = solve(A,b)
o9 = | -.15 |
| 1.1 |
| -.38 |
3 1
o9 : Matrix RR <--- RR
53 53
|
i10 : A*x-b
o10 = | -2.2e-16 |
| -2.2e-16 |
| 8.9e-16 |
3 1
o10 : Matrix RR <--- RR
53 53
|
i11 : norm oo
o11 = 8.88178419700125e-16
o11 : RR (of precision 53)
|
For large dense matrices over
RR or
CC, this function calls the lapack routines.
i12 : n = 10;
|
i13 : A = random(CC^n,CC^n)
o13 = | .21+.9i .91+.65i .3+.64i .96+.17i .043+.037i .12+.064i .038+.15i
| .05+.58i .26+.52i .21+.98i .34+.41i .34+.56i .092+.015i .82+.07i
| .59+.68i .57+.88i .12+.03i .82+.89i .77+.23i .73+.08i .95+.04i
| .23+.58i .42+.19i .39+.95i .96+.16i .29+.77i .15+.8i .067+.49i
| .2+.87i .51+.83i .5+.43i .97+.65i .47+.075i .17+.18i .83+.05i
| .06+.72i .71+.16i .94+.41i .32+.64i .88+.93i .45+.52i .11+.18i
| .81+.03i .095+.022i .11+.97i .67+.68i .88+.74i .25+.61i .96+.29i
| .88+.88i .95+.68i .97+.75i .67+.99i .32+.53i .68+.58i .47+.56i
| .45+.53i .63+.21i .93+.71i .87+.88i .53+.95i .89+.39i .61+.79i
| .29+.81i .88+.25i .75+.5i .35+.23i .23+.48i .46+.29i .24+.058i
-----------------------------------------------------------------------
.61+.57i .12+.65i .61+.54i |
.63+.32i .62+.58i .69+.65i |
.05+.55i 1+.85i .95+.16i |
.55+.35i .33+i .08+.97i |
.56+.21i .72+.76i .53+.03i |
.84+.57i .84+.18i .14+.64i |
.62+.86i .65+.33i .77+.34i |
.34+.67i .85+.34i .94+.11i |
.07+.66i .56+.58i .42+.46i |
.29+.38i .24+.79i .51+.72i |
10 10
o13 : Matrix CC <--- CC
53 53
|
i14 : b = random(CC^n,CC^2)
o14 = | .22+.71i .48+.11i |
| .059+.32i 1+.37i |
| .5+.32i .95+.2i |
| .74+.27i .8+.27i |
| 1+.87i .08+.6i |
| .011+.49i .64+.47i |
| .46+.63i .029+.14i |
| .69+.81i .84+.83i |
| .27+.21i .14+.4i |
| .43+.38i .11+.67i |
10 2
o14 : Matrix CC <--- CC
53 53
|
i15 : x = solve(A,b)
o15 = | .01+1.1i 4.4+2.8i |
| .72+.48i -.29-4.2i |
| .59+.48i .94-.03i |
| .17-.9i -.77-.44i |
| -.93+1.6i .98-1.2i |
| -.68-1.2i -1.1+3.5i |
| .94-.12i -2.1-.55i |
| .23+.21i .97+1.6i |
| .63-.52i .43-.65i |
| -.29-.86i -1.7-2.1i |
10 2
o15 : Matrix CC <--- CC
53 53
|
i16 : norm ( matrix A * matrix x - matrix b )
o16 = 1.54636173612489e-15
o16 : RR (of precision 53)
|
This may be used to invert a matrix over
ZZ/p,
RR or
QQ.
i17 : A = random(RR^5, RR^5)
o17 = | .41 .72 .11 .83 .45 |
| .57 .37 .91 .098 .19 |
| .54 .37 .84 .92 .12 |
| .83 .83 .63 .35 .45 |
| .48 .23 .8 .017 .16 |
5 5
o17 : Matrix RR <--- RR
53 53
|
i18 : I = id_(target A)
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 RR <--- RR
53 53
|
i19 : A' = solve(A,I)
o19 = | -4.2 -13 2.7 6.1 7.7 |
| .96 15 -1.5 -1.7 -14 |
| 1.5 5.4 -.83 -2.7 -2.1 |
| .28 -2 1.1 -.17 1.2 |
| 3.7 -9 -1.8 -1.9 13 |
5 5
o19 : Matrix RR <--- RR
53 53
|
i20 : norm(A*A' - I)
o20 = 3.5527136788005e-15
o20 : RR (of precision 53)
|
i21 : norm(A'*A - I)
o21 = 4.05231403988182e-15
o21 : RR (of precision 53)
|
Another method, which isn't generally as fast, and isn't as stable over
RR or
CC, is to lift the matrix
b along the matrix
A (see
Matrix // Matrix).
i22 : A'' = I // A
o22 = | -4.2 -13 2.7 6.1 7.7 |
| .96 15 -1.5 -1.7 -14 |
| 1.5 5.4 -.83 -2.7 -2.1 |
| .28 -2 1.1 -.17 1.2 |
| 3.7 -9 -1.8 -1.9 13 |
5 5
o22 : Matrix RR <--- RR
53 53
|
i23 : norm(A' - A'')
o23 = 0
o23 : RR (of precision 53)
|