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Every eigenvalue of a unitary matrix has absolute value 1. I was wondering whether a matrix whose eigenvalues all have absolute value 1 must be unitary?

Thanks!

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2 Answers 2

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  1. No, the eigenvectors of a unitary matrix must also be orthogonal. So for example the matrix with Eigenvectors (1,0) and (1,1) with eigenvalues 1 and -1, respectively, is not unitary.
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Thanks, but I think the eigenvectors of a unitary matrix are not necessarily orthogonal. If you are right, how would you characterize a unitary matrix via its eigenvalues and/or eigenvectors? –  Tim Nov 25 '12 at 19:48
    
At least their eigenspaces must be orhogonal so that we can choose an orthonormal set of eigenvectors, to be more precise. –  Dominik Nov 25 '12 at 19:50

2: Yes, if the algebraic multiplicity of all eigenvectors equal their geometric multiplicity, then the matrix is diagonalisable because the dimensions of the eigenspaces add up to $n$ so that you can choose $n$ linear independent eigenvectors (at least over an algebraically closed field)

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Thanks! I mean triangular, not necessarily diagonal. –  Tim Nov 25 '12 at 19:53
    
Is it possible to merge the answers? –  c.p. Nov 25 '12 at 19:59
    
@Dominik: I think there is some misunderstanding, is there? My question is about when matrices are triangular, and your reply is about when matrices are diagonalizable. I can't see the connection. (BTW, I am going to accept your other reply, and create a new post for the second question. You are welcome to move your reply there.) –  Tim Nov 26 '12 at 12:56
    
Diagonalizabilitiy is a special case of triangularizability. –  Dominik Nov 26 '12 at 13:03
    
@Dominik: I meant triangular, not triangularizability. –  Tim Nov 26 '12 at 14:48

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