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Math Help - Need help with determining Similarity of two matrices

  1. #1
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    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
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  2. #2
    A Plied Mathematician
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    Their determinants will have to be the same. I'm not sure if that's a sufficient condition, but it's certainly necessary.
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  3. #3
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    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.
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  4. #4
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    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:

    A=\begin{bmatrix}<br />
0 &-1 &1 &1\\<br />
-1 &1 &0 &0\\<br />
0 &0 &-1 &1\\<br />
0 &0 &0 &0<br />
\end{bmatrix},\quad\text{and}\quad<br />
B=\begin{bmatrix}<br />
-0.5 &-0.5 &-0.5 &-1.5\\<br />
-0.5 &1.5 &0.5 &-0.5\\<br />
0 &0 &-1 &1\\<br />
0 &0 &0 &0<br />
\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.
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  5. #5
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    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?
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  6. #6
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    Ah! I was able to find an S. I actually solved for it earlier when changing basis. Much appreciate your help Ackbeet.
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  7. #7
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    Note that both matrices are diagonalizable and similar to the same diagonal matrix:

    \begin{bmatrix}<br />
\frac{1+\sqrt{5}}{2} &0 &0 &0\\<br />
0 &\frac{1-\sqrt{5}}{2} &0 &0\\<br />
0 &0 &-1 &0\\<br />
0 &0 &0 &0<br />
\end{bmatrix}
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  8. #8
    A Plied Mathematician
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    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!
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