Solve a Matrix equation

Hi, I was struggling in this equation for serveal days.

A' * B * A =P

Solve A.

A is an unknown 4X1 complex matrix and A' is the transpose conjugate of A.

B is a known 4X4 complex matrix.

P is 1X1.

Do anybody know how to solve it on the matlab?

[Stephan]

Can you provide values for B and P?

Answer 1

[Stephan]

Symbolic approach:

A = sym('A', [4,1])
B = magic(4)
P = 13
sol = solve(A'*B*A==P, A)
sol.A1
sol.A2
sol.A3
sol.A4
% This example gives 4 solutiuons - we test the first one
test_A = [sol.A1(1); sol.A2(1); sol.A3(1); sol.A3(1)]
double(test_A' * B * test_A)

Answer 2

[Ameer Hamza]

There might be an algebric way to solve this problem, but following shows an optimization based method

rng(0);
B = rand(4)+1i*rand(4);
P = 0.5 + 0.3i;

fun = @(x) abs(x.'*B*x-P);
x0 = rand(8,1);
sol = fmincon(@(x) fun((x(1:4)+1i*x(5:8))), x0);

A = sol(1:4)+1i*sol(5:8);