Diagonalizing by Unitary Similarity Transformation

[Thadis]

Homework Statement

Compute the inverse, eigenvalues and eigenvectors of the following matrix, M.
Are the eigenvectors orthogonal? Determine a unitary similarity transformation
matrix U such that U^-1MU is diagonal.With M being

{2, 0, 2i, 0, 1}
{0, -1, 0,-2i,0}
{-2i, 0, 1, 1, 1}
{ 0, 2i, 1, 0,1}
{1, 0, 1, -1,-1}

Homework Equations

I know for that you are able to diagonalize real matrices by creating a matrix of the eigenvectors.

The Attempt at a Solution

I have tried solving this problem using Mathmatica by creating a matrix from the eigenvectors then inversing that matrix and using the U^-1MU identity to see if I get a diagonal matrix but I end up not getting a diagonal matrix. I also have tried orthogonalizing the eigenvector matrix to see if that was a problem but it did not seem to work. Does anyone have anything that might help me understand how to do this problem and also the logic behide the steps? Thank you!

 

[vela]

Homework Statement

Compute the inverse, eigenvalues and eigenvectors of the following matrix, M.
Are the eigenvectors orthogonal? Determine a unitary similarity transformation
matrix U such that U^-1MU is diagonal.With M being

{2, 0, 2i, 0, 1}
{0, -1, 0,-2i,0}
{-2i, 0, 1, 1, 1}
{ 0, 2i, 1, 0,1}
{1, 0, 1, -1,-1}

Homework Equations

I know for that you are able to diagonalize real matrices by creating a matrix of the eigenvectors.

The Attempt at a Solution

I have tried solving this problem using Mathmatica by creating a matrix from the eigenvectors then inversing that matrix and using the U^-1MU identity to see if I get a diagonal matrix but I end up not getting a diagonal matrix. I also have tried orthogonalizing the eigenvector matrix to see if that was a problem but it did not seem to work. Does anyone have anything that might help me understand how to do this problem and also the logic behide the steps? Thank you!

I notice that the matrix is nearly Hermitian. Do $m_{45}$ and $m_{54}$ have the right signs? In either case, it doesn't look like you get very nice answers for the eigenvalues from what I see from Mathematica.

 

[Dick]

I notice that the matrix is nearly Hermitian. Do $m_{45}$ and $m_{54}$ have the right signs? In either case, it doesn't look like you get very nice answers for the eigenvalues from what I see from Mathematica.

Yeah, I agree. I tried changing it both ways and I still don't get a characteristic polynomial I can factor or anything. But either way including the nonhermitian initial form, I still get 5 distinct eigenvalues. So with the right computer tools you should be able to get an approximate diagonalization. I know I don't have the right tools.

 

[Thadis]

I notice that the matrix is nearly Hermitian. Do $m_{45}$ and $m_{54}$ have the right signs? In either case, it doesn't look like you get very nice answers for the eigenvalues from what I see from Mathematica.

It is suppose to be a Hermitian matrix. They both should be -1, sorry about that. And I believe I have an approximate answer. I believed I just changed the way I did it and use the face that for a unility matrix that U-dagger =U^-1 and it was much cleaner output. Thanks for your help!