(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 = | .66+.76i .8+.97i .74+.1i .6+.04i .8+.66i .18+.26i .52+.51i
| .87+.94i .23+.59i .56+.93i .03+.38i .92+.01i .52+.64i .16+.6i
| .95+.44i .44+.26i .087+.25i .81+.39i .81+.54i .81+.33i .63+.12i
| .58+.42i .8+.32i .4+.56i .27+.16i .57+.37i .87+.34i .81+.69i
| .27+.44i .57+.14i .46+.84i .67i .14+.56i .42+.31i .54+.57i
| .57+.46i .5+.84i .061+.18i .36+.55i .78+.07i .75+.95i .28+.64i
| .52+.83i .65+.9i .22+.58i .53+.89i .058+.47i .32+.26i .87+.96i
| .62+.6i .74+.47i .64+.82i .43+.12i .91+.09i .93+.95i .53+.08i
| .91+.22i .81+.14i .5+.42i .3+.45i .48+.84i .79+.6i .98+.79i
| .99+.38i .28+.34i .57+.87i .99+.99i .43+.75i .25+.84i .1+.5i
-----------------------------------------------------------------------
.5+.58i .22+.53i .53+.75i |
.62+.83i .96+.14i .85+.77i |
.83+.47i .08+.98i .44+.78i |
.68+.21i .24+.75i .84+.74i |
.96+.65i .25+.58i .57+.33i |
.5+.15i .84+.61i .35+.39i |
.8+.65i .16+.88i .26+.33i |
.77+.42i .23+.91i .28+.76i |
.72+.04i .73+.68i .82+.48i |
.64+.66i .7+.35i .37+.26i |
10 10
o13 : Matrix CC <--- CC
53 53
|
i14 : b = random(CC^n,CC^2)
o14 = | .33+.49i .91+.55i |
| .74+.63i .94+.38i |
| .62+.54i .65+.79i |
| .22+.64i .64+.07i |
| .89+.18i .77+.42i |
| .13+.85i .78+.63i |
| .9+.59i .08+.53i |
| .15+.68i .72+.24i |
| .7+.06i .63+.98i |
| .43+.75i .96+.37i |
10 2
o14 : Matrix CC <--- CC
53 53
|
i15 : x = solve(A,b)
o15 = | 1.1-1.1i -.16+1.1i |
| -.17-.24i 1.1-.28i |
| -.65-.19i -.56i |
| -.49+1.6i -.42-1.3i |
| -.54-1.2i 1.6-.29i |
| 1.3+.63i -.29+.8i |
| -.36-.045i -.72+.05i |
| 1-.83i .22+.36i |
| .47+.098i -.03+.54i |
| -.54+2.1i -.39-.83i |
10 2
o15 : Matrix CC <--- CC
53 53
|
i16 : norm ( matrix A * matrix x - matrix b )
o16 = 1.11022302462516e-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 = | .27 .9 .43 .47 .2 |
| .49 .2 .1 .78 .14 |
| .14 .52 .89 .48 .22 |
| .18 .97 .052 .84 .22 |
| .071 .89 .094 .17 .38 |
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 = | 3.1 1.7 -1.5 -2.4 .0043 |
| 1.8 -.83 -.74 .28 -.35 |
| .4 -.27 1 -.48 -.42 |
| -1.7 .18 .75 1.6 -.56 |
| -4.1 1.6 1.4 -.8 3.8 |
5 5
o19 : Matrix RR <--- RR
53 53
|
i20 : norm(A*A' - I)
o20 = 2.22044604925031e-16
o20 : RR (of precision 53)
|
i21 : norm(A'*A - I)
o21 = 4.44089209850063e-16
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 = | 3.1 1.7 -1.5 -2.4 .0043 |
| 1.8 -.83 -.74 .28 -.35 |
| .4 -.27 1 -.48 -.42 |
| -1.7 .18 .75 1.6 -.56 |
| -4.1 1.6 1.4 -.8 3.8 |
5 5
o22 : Matrix RR <--- RR
53 53
|
i23 : norm(A' - A'')
o23 = 0
o23 : RR (of precision 53)
|