# Thread: Need help with determining Similarity of two matrices

1. ## Need help with determining Similarity of two matrices

Hey everybody,

Given two 4x4 Matrices
A = [0 -1 1 1, -1 1 0 0, 0 0 -1 1, 0 0 0 0] B = [-0.5 -0.5 -0.5 -1.5, -0.5 1.5 0.5 -0.5, 0 0 -1 1, 0 0 0 0]

I need to show that these two matrices are similar. I understand that I need to find a non-singular invertible matrix S that satisfies the equation: AS = SB, but I have searched everywhere and have no clue as to how to find this matrix for 4x4 matrices. I have not learned eigenvalues or eigenvectors yet.

Thanks for any responses

2. Their determinants will have to be the same. I'm not sure if that's a sufficient condition, but it's certainly necessary.

3. Thank you for a quick response Ackbeet.

However, I need to find a sufficient condition to show that the two matrices are indeed similar. I am having major difficulties trying to find the invertible matrix S that would show the two matrices are similar.

4. Hmm. I would totally solve this problem using eigenvalues and eigenvectors. Otherwise, you'd need to set up a system of 16 variables and 16 unknowns, which would take a while to solve, unless you're allowed to use a computer/calculator. Obviously, if you can compute the matrix S, and show that it has a nonzero determinant, then by definition you'd have shown the result you're after. So, just to clarify, you need to show that the following two matrices are similar:

$\displaystyle A=\begin{bmatrix} 0 &-1 &1 &1\\ -1 &1 &0 &0\\ 0 &0 &-1 &1\\ 0 &0 &0 &0 \end{bmatrix},\quad\text{and}\quad B=\begin{bmatrix} -0.5 &-0.5 &-0.5 &-1.5\\ -0.5 &1.5 &0.5 &-0.5\\ 0 &0 &-1 &1\\ 0 &0 &0 &0 \end{bmatrix}.$

Is this correct?

Without using eigenvalues, I think you'll have to set up the system and solve it on a computer. I was able to do that, and I did find an S that works.

5. Yup, those are the matrices that I need to find. Sorry I did not know how to display the matrices like you have just done.

Is it much more straightforward to solve with eigenvalues? I am trying to do the problem by hand so I am trying to avoid computer programs as much as possible. *EDIT*Would I need matlab or similar program to solve this on computer?

6. Ah! I was able to find an S. I actually solved for it earlier when changing basis. Much appreciate your help Ackbeet.

7. Note that both matrices are diagonalizable and similar to the same diagonal matrix:

$\displaystyle \begin{bmatrix} \frac{1+\sqrt{5}}{2} &0 &0 &0\\ 0 &\frac{1-\sqrt{5}}{2} &0 &0\\ 0 &0 &-1 &0\\ 0 &0 &0 &0 \end{bmatrix}$

8. With eigenvalues and eigenvectors, the problem is quite straightforward. After you find your eigenvalues, and your (hopefully n linearly independent) eigenvectors, you form the matrix S out of the eigenvectors. Done.

Glad you solved it. Have a good one!