Problem finding 2 unknown matrices in 2 equations

Hello,

I have 2 equations containing 2 unknown matrices of size nxn that I need to find. The equations looks like this.

I+R=E*T
B1-B1*R=E*B2*T

Where I is the Identity matrix og size nxn, and B1, B2 and E are known matrices of size nxn.

It seems like the built in solve functions doesnt work with matrices, and I cant seem to figure out how to solve it. Can anyone help with this?

Edit: I forgot to mention that the matrices B1 and B2 are diagonal matrices.

- Nicolai Hagelund

[Matt J]

How large is n?

[Nicolai Hagelund]

n is 998 in the current case, but can vary depending on earlier code.

Answer 1

[Matt J]

If you multiply the first equation by B1 and add the equations, you will eliminate R

 2*B1=(B1*E+E*B2)*T

You can then just solve for T linear algebraically (backslash) and then use equation (1) to obtain R.

[Nicolai Hagelund]

I cannot see how you can do that since multiplying the first equation gives(in my mind)

B1+B1*R=B1*E*T

And the other equation is

B1-B1*R=E*B2*T

Having a negative B1*R, meaning that is not viable, unless I'm missing something.

Sorry for the long wait on your comment, some other stuff came along that I had to give priority.

[Matt J]

Yes, you now have 2 equations

B1+B1*R=B1*E*T
B1-B1*R=E*B2*T

If you add them together, the +B1*R and -B1*R terms cancel with each other leading to the equation I showed you before

2*B1=(B1*E+E*B2)*T

Now, R has been eliminated and you just solve this for T.

[Nicolai Hagelund]

Oh, I thought you meant something else. I sse what you mean now.

Thanks, great help, and exactly what I needed.

Answer 2

[Matt J]

If n isn't too large,

 I=speye(n);

 A=[kron(I,I), kron(-I,E); kron(I, B1), kron(I,E*B2)]
 b =[-I(:);B1(:)];
 RT=full(reshape(A\b,n,2*n));

 R=RT(1:n,1:n);
 T=RT(1:n, n+1:end);

[Nicolai Hagelund]

n is quite large, atleast 998 and can vary. I forgot to mention in the question that the B1 and B2 are diagonal matrices.