(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 = | .55+.16i .85+.16i .08+.8i .63+.18i .93+.74i .86+.87i .04+.43i
| .13+.71i .4+.77i .25+.98i .79+.74i .3+.71i .7+.99i .77+.38i
| .25+.33i .92+.65i .047+.026i .15+.26i .38+.19i .6+.91i .33+.16i
| .88+.59i .68+.87i .43+.76i .35+.91i .14+.87i .1+.52i .14+.1i
| .29+.7i .97+.95i .6+.22i .23+.44i .04+.57i .54+.46i .1+.46i
| .1+.6i .66+.73i .72+.26i .78+.63i .43+.36i .3+.014i .84+.22i
| .32+.61i .74+.57i .82+.51i .23+.36i .12+.47i .77+.19i .86+.75i
| .8+.95i .31+.66i .44+.098i .09+.91i .032+.21i .019+.18i .94+.58i
| .55+.73i .26+.91i .26+.75i .28+.36i .92+.93i .8+.32i .42+.46i
| .41+.45i .62+.15i .77+.75i .077+.24i .26+.76i .77+.69i .19+.3i
-----------------------------------------------------------------------
.71+.8i .02+.34i .91+.72i |
.53+.4i .46+.33i .09+.55i |
.58+.16i .75+.14i .03+.56i |
.8+.89i .36+.13i .15+.51i |
.48+.34i .61+.55i .62+.09i |
.84+.99i .92+.7i .25+.69i |
.76+.54i .63+.98i .12+.57i |
.4+.15i .2+.98i .78+.51i |
.03+.94i .63+.73i .47+.088i |
.93+.89i .4+.043i .065+.21i |
10 10
o13 : Matrix CC <--- CC
53 53
|
i14 : b = random(CC^n,CC^2)
o14 = | .34+.63i .6+.8i |
| .58+.97i .49+.54i |
| .64+.24i .73+.04i |
| .18+.97i .7+.37i |
| .92+.98i .2+.41i |
| .83+.89i .19+.87i |
| .62+.06i .78+.51i |
| .71+.93i .91+.82i |
| .07+.69i .06+.62i |
| .56+.78i .1+.74i |
10 2
o14 : Matrix CC <--- CC
53 53
|
i15 : x = solve(A,b)
o15 = | .26-.61i -.21-.29i |
| .05+.0067i 1.4-.33i |
| .15+1.2i -1.5-.51i |
| 1-.11i -.88+.13i |
| -.45+.58i .45-.59i |
| .74-.52i .005+.45i |
| -.016-.17i 1+2i |
| .29-.52i 1.8+.76i |
| -.49-.097i -1.5-1.7i |
| -.18+.67i -.57+.79i |
10 2
o15 : Matrix CC <--- CC
53 53
|
i16 : norm ( matrix A * matrix x - matrix b )
o16 = 6.66133814775094e-16
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 = | .28 .86 .039 .077 .58 |
| .077 .41 .55 .82 .0023 |
| .055 .4 .046 .98 .65 |
| .33 .21 .076 .7 .83 |
| .17 .59 .41 .78 .63 |
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 = | 1.6 3 -1.6 3.6 -4.6 |
| 1.3 .44 .71 -.99 -.6 |
| -1.2 -.63 -2 -.41 3.7 |
| .0074 1.1 1.1 .42 -1.7 |
| -.85 -2.2 -.33 -.31 3.1 |
5 5
o19 : Matrix RR <--- RR
53 53
|
i20 : norm(A*A' - I)
o20 = 2.77555756156289e-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 = | 1.6 3 -1.6 3.6 -4.6 |
| 1.3 .44 .71 -.99 -.6 |
| -1.2 -.63 -2 -.41 3.7 |
| .0074 1.1 1.1 .42 -1.7 |
| -.85 -2.2 -.33 -.31 3.1 |
5 5
o22 : Matrix RR <--- RR
53 53
|
i23 : norm(A' - A'')
o23 = 0
o23 : RR (of precision 53)
|