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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!

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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    At least their eigenspaces must be orhogonal so that we can choose an orthonormal set of eigenvectors, to be more precise.2012-11-25
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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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    Then the answer must be no, of course. For example you can consider {{1,2},{3,4}}^-1*{{1,0},{0,-1}}*{{1,2},{3,4}}={{-5,8}{3,5}}.2012-11-26