Reformulate a matrix equation

Greetings everyone. I have a question: I have 3 matrices which contain numeral values, all of them are of the same order. The matrices are A,J and S.

The equation of is J = inv(S) * A * S;

Is there any way possible for me to solve this for S? i.e. J and A are known but S is unknown, can I do anything to obtain S from J and A? Thank you in advance.

Regards, M. Ayoub

*Edit: S is unknown, when I said it contains numeral values I meant that that unknowns are of numeral type (i.e. not string or char, etc...).

*Edit: all of the matrices are square matrices.

[Roger Stafford]

If it is assumed that an inverse exists for S, then your equation implies that S*J-A*S = 0. For n x n matrices, this is consequently n^2 linear equations in n^2 unknown S values. However, I think you will find that unless J and A bear a very special relation to one another, the associated n^2 x n^2 matrix of J and A coefficients has rank n^2, and this latter equation will have only the trivial solution S = 0. S would therefore possess no inverse. Consequently, in the general case there is no solution to your equation. That is, of course, not a fault of Matlab. It is a general mathematical property.

Answer 1

[Walter Roberson]

If J = inv(S)*A*S then J*inv(S) = inv(S)*A*S*inv(S) so J*inv(S) = inv(S) * A. Then S*J*inv(S) = S*inv(S) * A, so S*J*inv(S) = A and so

A = S*J*inv(S)

Your question assumes that S is invertable, so as long as you have the numeric values for S and J and S is invertable then you can calculate A.

Note: sometimes people ask this question when the matrices are not square. That case cannot usually be solved.

[Mohammad Ayoub]

They’re actually all square. I’ll put that in my question thank you. As for your answer, I got it. But my main concern here is to calculate S. Is there any way possible? I tried

S = syms(‘S’, [4 4]) %A and J are of order 4

solve(S*J == A*S,S)

But it gives me weird outputs...

[M]

I think this is not that simple, as the question was to calculate S, with known matrices A and J.

But you can use nonlinear matrix solver to find the value of S.

[Mohammad Ayoub]

Is there a way to solve nonlinear matrix equations using MATLAB?

[M]

One possible way is to solve fsolve.

[Walter Roberson]

Ah, unfortunately the solve(S*J==A*S,S) formulation gives S as all zero, which is a solution when written in that form...)

[Walter Roberson]

On the test random data I used, vpasolve(J == inv(S)*A*S) said there was no solution.

[Mohammad Ayoub]

Oh well. I guess I can’t achieve that using MATLAB. Thank you very much! Much appreciated

[Walter Roberson]

It might have worked on other data.