Solve this matrix equation and find T

I would like to find a matrix T that satisfies this equation

T^-1*F*T = G

where F and G are known matrices.

In particular,

G = [0 2 0;-1 1 0;-2 2 0]

and

F = [0 -1 0; 2  1 0; 0 1 0]

Thanks!

Answer 1

[John D'Errico]

Homework, right? Why not admit it?

Note that the solution, if one exists, is not unique, since IF T does exist to solve the problem, then it is also true that for any scalar k, we also have k*T as a fully valid solution.

If T exists, such that T^-1*F*T = G, then we can write the problem in this form:

F*T = T*G

Now solve it using kron and null. The trick is to "unwind" the matrix T into a vector. Use kron to implicitly do that.

G = [0 2 0;-1 1 0;-2 2 0];
F = [0 -1 0; 2  1 0; 0 1 0];

sol = null(kron(eye(3),F) - kron(G',eye(3)));

T = reshape(sol(:,1),[3,3])
T =
     -0.44989      0.17925   1.1936e-16
      0.17925      0.72053  -4.7178e-17
      0.44989     -0.10133    -0.038959

I picked the first column of sol, but that choice was arbitrary. I could have used any linear combination of the columns of sol. Regardless, did it work?

inv(T)*F*T
ans =
   3.8858e-16            2   9.5242e-19
           -1            1   2.6559e-16
           -2            2   5.3118e-16
           
norm(inv(T)*F*T - G)
ans =
   1.0165e-14

So, ignoring the stuff that is on the order of eps, T does as required.