solve system of matrices

% I want to solve this system
f'(A,B)*[dA dB]'=[g(C))+B   B*A-eye(n-1)]'

but when I found the derivative of f in the left side, I found that

f'(A,B)*[dA dB]'= [g(dA))-dB   A*dB+B*dA]'

where A,B are (n-1)×(n-1) matrices

and C is n×n matrices

and I define a function chi from (n-1)×(n-1) matrices to the n×n matrices as

function chi= g(X)
chi= 3*trace X;
end

I want to solve this system for dA and dB

the problem for me is in the derivative part here

f'(A,B)*[dA dB]'= [g(dA))-dB   A*dB+B*dA]'

can I said

[g(dA))-dB   A*dB+B*dA]'= [g  -eye(n-1);B    A]*[dA   dB]'

if not can I solve this system for dA and dB if i write it as

[g(dA)-dB  A*dB+B*dA]'=[g(C)+B   B*A-eye(n-1)]'

[Torsten]

As far as I understand your question, you want - given A, B and C - determine dA and dB both matrices of size

(n-1) x (n-1) such that

[g(dA)-dB  A*dB+B*dA]'=[g(C)+B   B*A-eye(n-1)]'

where g is defined as

g(M) = 3*trace(M)

A,B are matrices of size (n-1) x (n-1) and C is a matrix of size n x n.

Is this correct ?

[Hajar Alshaikh]

Yes it is correct except that C is n-1×n-1 matrix I wrote the size of C as n×n by mistake

[Torsten]

Then transform your matrix equation to the form

M*x = b

by using MATLAB's "equationsToMatrix" and solve for the unknown matrices dA and dB using

x = M\b.

[Hajar Alshaikh]

this way is not working because we deal here with matrices and in your way the variables have to be x and b are vectors

[Torsten]

The matrix equation you have is in reality a linear system of equations in the matrix entries of dA and dB.

"equationsToMatrix" transforms your matrix equation to this underlying linear system.

After getting the solution x of this linear system, you will have to reshape it back to get the matrices dA and dB.

[Hajar Alshaikh]

I read the examples about "equationsToMatrix" and all deal with b as vector and in my situation the problem is that all A, x, and b are matrices and the other problem is that I cannot seprate the component in x than the matrix A .

I mean than we cannot say

[g(dA))-dB   A*dB+B*dA]'= [g  -eye(n-1);B    A]*[dA   dB]'

[Torsten]

So this is not what you want ?

rng("default")
n = 20;
A = rand(n-1);
B = rand(n-1);
C = rand(n-1);
dA = sym('dA',[n-1 n-1],'real');
dB = sym('dB',[n-1 n-1],'real'); 
eqn = [3*trace(dA)-dB  A*dB+B*dA]'==[3*trace(C)+B   B*A-eye(n-1)]';
eqn = eqn(:);
vars = [reshape(dA.',[(n-1)^2 1]);reshape(dB.',[(n-1)^2 1])];
[M,b] = equationsToMatrix(eqn,vars);
sol = double(M\b);
dA = reshape(sol(1:(n-1)^2),[n-1 n-1]).';
dB = reshape(sol((n-1)^2+1:2*(n-1)^2),[n-1 n-1]).';
[3*trace(dA)-dB  A*dB+B*dA]'-[3*trace(C)+B   B*A-eye(n-1)]'

ans = 38×19

1.0e-14 * -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553 -0.3553

[Hajar Alshaikh]

first of all I really want to thank you about your help and i appritiate your time, I learend from you many functions that I dont know about them before.

Now I tried your way, but I got this error

Warning: Solution does not exist because the system is inconsistent.
Error using vertcat
Dimensions of arrays being concatenated are not consistent.

[Torsten]

My example worked - so I cannot tell you what went wrong with your code.

Answer 1

[Torsten]

Should be faster than the symbolic solution.

rng("default")
n = 50;
A = rand(n-1);
B = rand(n-1);
C = rand(n-1);
x0 = zeros(2*(n-1)^2,1);
x = fsolve(@(x)fun(x,A,B,C,n),x0);

Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.

dA = reshape(x(1:(n-1)^2),[n-1,n-1])

dA = 49×49

1.0e+03 * -0.9697 -1.2841 -1.1100 -1.0445 -1.1586 -1.2281 -1.0768 -0.9605 -1.0724 -1.0369 -1.1764 -0.9773 -0.9512 -1.0230 -1.1881 -0.9808 -1.0192 -1.0678 -1.2432 -1.0212 -1.0383 -0.9359 -0.9486 -1.1359 -1.0584 -1.1338 -1.1257 -0.9812 -0.9628 -0.9208 -0.5211 -0.6925 -0.5981 -0.5635 -0.6226 -0.6578 -0.5814 -0.5162 -0.5780 -0.5564 -0.6303 -0.5278 -0.5112 -0.5491 -0.6402 -0.5259 -0.5471 -0.5732 -0.6705 -0.5500 -0.5594 -0.5001 -0.5119 -0.6114 -0.5700 -0.6076 -0.6053 -0.5272 -0.5172 -0.4967 0.2252 0.2944 0.2581 0.2441 0.2652 0.2793 0.2498 0.2234 0.2489 0.2389 0.2685 0.2307 0.2179 0.2377 0.2720 0.2269 0.2369 0.2441 0.2848 0.2372 0.2407 0.2138 0.2232 0.2611 0.2435 0.2586 0.2588 0.2268 0.2248 0.2153 0.2900 0.3845 0.3341 0.3153 0.3454 0.3672 0.3232 0.2882 0.3204 0.3103 0.3511 0.2943 0.2830 0.3059 0.3565 0.2920 0.3050 0.3196 0.3719 0.3056 0.3105 0.2781 0.2845 0.3404 0.3153 0.3384 0.3377 0.2935 0.2883 0.2761 0.9613 1.2734 1.1008 1.0342 1.1489 1.2161 1.0665 0.9516 1.0624 1.0265 1.1667 0.9681 0.9430 1.0141 1.1791 0.9709 1.0102 1.0580 1.2321 1.0116 1.0289 0.9269 0.9393 1.1246 1.0487 1.1223 1.1157 0.9738 0.9525 0.9124 0.4635 0.6062 0.5273 0.4976 0.5500 0.5843 0.5120 0.4595 0.5122 0.4952 0.5603 0.4662 0.4534 0.4860 0.5625 0.4661 0.4895 0.5086 0.5910 0.4872 0.4929 0.4485 0.4539 0.5406 0.5028 0.5393 0.5354 0.4676 0.4623 0.4404 0.3057 0.4060 0.3501 0.3310 0.3651 0.3872 0.3408 0.3036 0.3384 0.3259 0.3707 0.3096 0.3003 0.3228 0.3775 0.3085 0.3202 0.3362 0.3932 0.3226 0.3280 0.2935 0.2991 0.3589 0.3339 0.3559 0.3557 0.3093 0.3034 0.2922 0.6562 0.8683 0.7499 0.7020 0.7852 0.8329 0.7274 0.6510 0.7249 0.7032 0.8006 0.6565 0.6466 0.6907 0.8043 0.6631 0.6925 0.7231 0.8428 0.6889 0.7014 0.6374 0.6407 0.7674 0.7176 0.7690 0.7614 0.6654 0.6508 0.6224 1.4197 1.8876 1.6310 1.5331 1.6967 1.7968 1.5786 1.4065 1.5696 1.5174 1.7234 1.4345 1.3923 1.5008 1.7445 1.4359 1.4941 1.5624 1.8221 1.4944 1.5230 1.3674 1.3917 1.6608 1.5518 1.6598 1.6481 1.4376 1.4084 1.3491 -0.3479 -0.4612 -0.3958 -0.3722 -0.4170 -0.4424 -0.3865 -0.3433 -0.3847 -0.3716 -0.4258 -0.3469 -0.3432 -0.3660 -0.4280 -0.3510 -0.3667 -0.3843 -0.4484 -0.3650 -0.3720 -0.3375 -0.3386 -0.4079 -0.3806 -0.4080 -0.4035 -0.3528 -0.3436 -0.3303

dB = reshape(x((n-1)^2+1:2*(n-1)^2),[n-1,n-1])

dB = 49×49

-0.7671 -1.4794 -1.1889 -0.6632 -0.6091 -0.9174 -0.9572 -0.8564 -1.2919 -1.5003 -1.4821 -1.1780 -1.2001 -1.4399 -0.9749 -1.0941 -1.3005 -1.1523 -1.2406 -0.8458 -0.8755 -1.2904 -1.1953 -0.7815 -1.1088 -0.8290 -0.7059 -1.4871 -1.0271 -0.7387 -0.7014 -0.9972 -1.4334 -0.6450 -1.4131 -0.8650 -1.0542 -0.9570 -0.8857 -1.1241 -1.2580 -0.6627 -1.5752 -1.0206 -1.0571 -0.5852 -1.2458 -1.4416 -1.3939 -1.4102 -0.7027 -1.0503 -0.7914 -1.5710 -0.5791 -0.6178 -0.8306 -1.5393 -0.8213 -1.0100 -1.3987 -0.9358 -1.2485 -1.4661 -1.4134 -0.7486 -1.4896 -1.4423 -1.2488 -1.0516 -0.9565 -1.5000 -1.0950 -1.1257 -1.1427 -1.2666 -0.7565 -0.7762 -1.3715 -1.5730 -0.9660 -1.2863 -1.2233 -0.6713 -1.4615 -0.8110 -1.4616 -1.1483 -1.4464 -1.0828 -1.2156 -1.0667 -1.1013 -0.8108 -0.6275 -0.9769 -0.5925 -0.9976 -1.2301 -1.0742 -1.2096 -0.6313 -1.5682 -1.1445 -1.0673 -1.5237 -1.1282 -1.2501 -1.0468 -1.2274 -1.3954 -1.5357 -0.9583 -1.2283 -0.9820 -0.9388 -0.7739 -1.1405 -1.1063 -0.9530 -0.5938 -0.8336 -0.8765 -1.4393 -1.1236 -1.2753 -0.7344 -0.8175 -1.1087 -0.8866 -0.8209 -1.1047 -0.8042 -1.2581 -0.8475 -1.4512 -1.5375 -1.4795 -0.8872 -1.2816 -1.5588 -1.0834 -0.6814 -0.7928 -0.8789 -1.2111 -0.6990 -0.7543 -1.4918 -1.0580 -1.4736 -1.5068 -1.2816 -1.2894 -1.5208 -0.7813 -1.0492 -1.1753 -1.2928 -1.5285 -1.1491 -0.6965 -0.9757 -0.9490 -1.5674 -0.6909 -1.1737 -0.7768 -1.2652 -1.5100 -1.4397 -0.8827 -0.9552 -0.8215 -1.5282 -1.5638 -1.1214 -1.0913 -1.5516 -0.9201 -1.0930 -1.0444 -0.9593 -1.4505 -0.8991 -1.2440 -1.1207 -1.0571 -1.0825 -1.5597 -1.5594 -0.9578 -1.2742 -0.6559 -0.7613 -0.9322 -1.3862 -0.8759 -1.5645 -1.2653 -0.6615 -1.3675 -0.8405 -0.9174 -1.0383 -0.7848 -0.8923 -1.1261 -1.1631 -1.3548 -1.1222 -0.8317 -1.1453 -1.5157 -1.3841 -1.0207 -0.6374 -1.4762 -1.0657 -1.0912 -1.4273 -1.3905 -0.6423 -1.0340 -1.4393 -0.8196 -1.5622 -1.0742 -1.3476 -1.1460 -0.9154 -0.8141 -0.8190 -0.7755 -0.8653 -1.3347 -0.9597 -0.7429 -0.6966 -0.9616 -1.1841 -1.0089 -1.4655 -0.7174 -1.1791 -1.0110 -1.2357 -1.5124 -1.0755 -1.5703 -0.8611 -0.8218 -1.3253 -0.6255 -0.6103 -1.1380 -1.4636 -1.4676 -1.4072 -0.9585 -0.8138 -0.8120 -1.2006 -1.0845 -0.6623 -1.4640 -1.3692 -1.0716 -1.5042 -1.2892 -1.3381 -1.2802 -1.4206 -0.9716 -1.3673 -0.7529 -1.4673 -1.3956 -1.5136 -1.0335 -1.2601 -1.4621 -0.9981 -1.3159 -0.9096 -1.1904 -0.7915 -1.0791 -1.2837 -1.2122 -0.8955 -1.0424 -1.1006 -1.5284 -1.1598 -1.0499 -1.4168 -1.1128 -1.1712 -1.0586

[3*trace(dA)-dB  A*dB+B*dA]'-[3*trace(C)+B   B*A-eye(n-1)]'

ans = 98×49

1.0e-10 * 0.2774 0.2773 0.2773 0.2773 0.2774 0.2774 0.2774 0.2773 0.2774 0.2774 0.2773 0.2773 0.2773 0.2774 0.2774 0.2774 0.2774 0.2774 0.2774 0.2774 0.2773 0.2773 0.2773 0.2773 0.2773 0.2773 0.2773 0.2773 0.2774 0.2774 0.2774 0.2774 0.2774 0.2773 0.2774 0.2773 0.2774 0.2774 0.2773 0.2773 0.2774 0.2774 0.2774 0.2773 0.2773 0.2773 0.2774 0.2774 0.2773 0.2773 0.2774 0.2773 0.2773 0.2774 0.2773 0.2774 0.2774 0.2773 0.2774 0.2774 0.2774 0.2774 0.2773 0.2774 0.2773 0.2774 0.2773 0.2774 0.2773 0.2774 0.2773 0.2774 0.2773 0.2774 0.2774 0.2774 0.2774 0.2774 0.2774 0.2774 0.2773 0.2773 0.2774 0.2774 0.2774 0.2774 0.2773 0.2773 0.2774 0.2774 0.2774 0.2773 0.2774 0.2774 0.2774 0.2773 0.2774 0.2773 0.2773 0.2773 0.2773 0.2774 0.2774 0.2773 0.2774 0.2773 0.2774 0.2774 0.2774 0.2774 0.2773 0.2773 0.2774 0.2774 0.2774 0.2774 0.2774 0.2773 0.2773 0.2774 0.2773 0.2774 0.2774 0.2774 0.2773 0.2774 0.2773 0.2773 0.2774 0.2774 0.2774 0.2774 0.2773 0.2774 0.2773 0.2773 0.2773 0.2774 0.2774 0.2773 0.2773 0.2775 0.2773 0.2773 0.2774 0.2774 0.2773 0.2773 0.2773 0.2774 0.2773 0.2773 0.2773 0.2774 0.2774 0.2773 0.2774 0.2773 0.2773 0.2773 0.2774 0.2774 0.2774 0.2774 0.2774 0.2773 0.2773 0.2774 0.2774 0.2773 0.2773 0.2774 0.2774 0.2774 0.2773 0.2774 0.2774 0.2774 0.2773 0.2774 0.2774 0.2773 0.2774 0.2774 0.2774 0.2773 0.2774 0.2774 0.2774 0.2774 0.2774 0.2774 0.2774 0.2773 0.2774 0.2774 0.2774 0.2773 0.2774 0.2774 0.2773 0.2774 0.2774 0.2774 0.2774 0.2774 0.2773 0.2773 0.2774 0.2773 0.2774 0.2774 0.2774 0.2773 0.2773 0.2773 0.2774 0.2774 0.2773 0.2773 0.2773 0.2774 0.2774 0.2773 0.2773 0.2774 0.2774 0.2774 0.2773 0.2774 0.2773 0.2773 0.2774 0.2774 0.2773 0.2773 0.2774 0.2774 0.2774 0.2774 0.2773 0.2774 0.2774 0.2774 0.2774 0.2773 0.2774 0.2774 0.2773 0.2773 0.2773 0.2774 0.2774 0.2774 0.2774 0.2773 0.2773 0.2774 0.2774 0.2774 0.2773 0.2773 0.2774 0.2773 0.2774 0.2774 0.2773 0.2773 0.2774 0.2773 0.2774 0.2773 0.2774 0.2774 0.2774 0.2774 0.2773 0.2773 0.2774 0.2773 0.2773 0.2774 0.2774 0.2774 0.2774 0.2773 0.2774 0.2774 0.2773 0.2774 0.2773 0.2774 0.2774 0.2773 0.2775 0.2774 0.2774 0.2773 0.2773 0.2774

function res = fun(x,A,B,C,n)
  dA = reshape(x(1:(n-1)^2),[n-1,n-1]);
  dB = reshape(x((n-1)^2+1:2*(n-1)^2),[n-1,n-1]);
  res = [3*trace(dA)-dB  A*dB+B*dA]' - [3*trace(C)+B   B*A-eye(n-1)]';
  res = res(:);
end