Finding Congruencing Matrix between two given matrices

I have two given nxn matrices A,M and I am trying to find P such that

P*A*P.'=M.

How can I find the solution for P in matlab?

Answer 1

[Torsten]

https://de.mathworks.com/matlabcentral/answers/278877-how-to-findthe-permutation-matrices-that-produce-another-matrix

[Roi Binia]

M isn't a permutation of A.

[Torsten]

So P can be arbitrary ? Usually P stands for a permutation matrix, i.e. a permutation of the rows/columns of eye(n). Or is P a unitary matrix ?

[Roi Binia]

P is just in GL_7

[Torsten]

And you want to decide whether A and M are congruent or you know it ?

[Roi Binia]

I know it, I am trying to find the congruencing matrix.

[Torsten]

Do you know of standard approaches from the literature ? I don't. Seems it is a hard problem with no ready-to-use solver.

Answer 2

[Bruno Luong]

Partial solution:

if A and M are real symmetric or complex Hemitian, the P can be computed as

[W,E]=eig(M,'vector');
[V,D]=eig(A,'vector');
P = W*diag(sqrt(E./D))*V';

Note that if A and M real,

• (A+A') and (M+M') are pure real symmetric,
• 1i*(A-A') and 1i*(M-M') are pure imaginary Hermitian
• Any real combination of both are Hermitian

[Torsten]

Note that if A and M real,

• (A+A') and (M+M') are pure real symmetric,
• 1i*(A-A') and 1i*(M-M') are pure imaginary Hermitian
• Any real combination of both are Hermitian

Do you think this helps for the general case A, M in GL(7,IR) or why did you add this ?

[Bruno Luong]

If I know I would post the solution for general case.