How to obtain the value of the matrix X in equation AXA' = B?

If I have

A*X*A' = B

X and B are real square matrices.

A might not be square matrix.

For example:

A (3x4) is given

B (3x3) is given

X (4x4) is unknown matrix.

How can I obtain the X matrix?

Answer 1

[KSSV]

For the give dimensions and given equation it cannot be solved. But for given dimensions the following equation can be solved.

% A'*X*A = B

To solve above A can be (3x4) , X is (3,3) abd B is (4x4).

A = rand(3,4) ;
B = rand(4,4) ;
X = pinv(A')*B*pinv(A)

X = 3×3

8.6286 7.7981 -10.7310 8.3291 3.1179 -8.3506 -8.9399 -3.2050 10.3068

If you want to solve the given equation:

% A*X*A' = B

The dimensions of A should be (4x3), X is (3x3) and B is (4X4)

A = rand(4,3) ;
B = rand(4,4) ;
X = pinv(A)*B*pinv(A')

X = 3×3

1.7788 -1.3116 1.1470 1.2149 -0.0710 -0.6240 -0.8625 1.7781 -0.6461

Check the dimensions once for your quesion. Read about pinv function and it's properties.

[Muhammad Rony Hidayatullah]

Thank you very much for the clever solution.

Answer 2

[Torsten]

If you multiply out A*X*A' for a matrix of unknowns X=(x_ij) and compare the elements with those of the matrix B, you get an (in your case) underdetermined linear system of equations for the x_ij. This usually has an infinite number of solutions.

[Muhammad Rony Hidayatullah]

thanks for your explanation