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If $A$ is an $n\times n$-matrix and it has $n$ orthonormal eigenvectors,

is it true that $U^*AU$ is diagonal? $U$ is an unitary matrix and $U^*$ is the conjugate transpose of $U$.

If it is true, how to prove it?


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I'm not too sure what you mean here. If $A$ has $n$ orthonormal eigenvectors, then almost by definition it is unitarily diagonalizable. What does this have to do with the Schur canonical form (by which I assume you mean the Schur triangularization)? – EuYu Nov 29 '12 at 17:24
The OP probably meant the columns of $U$ are the eigenvectors of $A$. – user1551 Nov 29 '12 at 17:29
For some basic information about writing math at this site see e.g. here, here, here and here. – Julian Kuelshammer Nov 29 '12 at 17:29
@EuYu thanks for your reply. Could you give me some links that talk about the unitarily diagonalizable? Especially why A has n orthonormal eigenvectors then it is unitarily diagonalizable. – mitchelllc Nov 29 '12 at 17:37
up vote 0 down vote accepted

Unitary diagonalization is really the same thing as normal diagonalization, except the eigenvectors form an orthonormal basis instead of a regular basis.

Recall that you diagonalize by forming an eigenbasis $\mathcal{B}$ of some matrix $A$. If the matrix $P$ has columns $\mathcal{B}$ then $P^{-1}AP$ will be diagonal. Now if the basis you took was orthonormal, then it follows that $P$ is not only invertible, but unitary. You have unitarily diagonalized $A$.

Of course, this is not always possible. Not all matrices are diagonalizable, and of those that are, still not all are unitarily diagonalizable. But you are given that your matrix has an orthonormal set of $n$ eigenvectors which by the above discussion is equivalent to the matrix being unitarily diagonalizable. The condition of being unitarily diagonalizable is equivalent to that of being normal in the sense of $AA^* = A^*A$.

Now Schur decomposition is something different, albeit related. Over $\mathcal{C}$ there always exists a unitary matrix $U$ for which $U^*AU = T$ is upper triangular. This is what is typically called the Schur decomposition or Schur triangularization. I assume this is what you mean by "Schur canonical form". It turns out that if your matrix $A$ is normal, then the unitary matrix which triangularizes $A$ also diagonalizes $A$. I suspect this is what your question is really asking. In fact this is the standard proof given that a normal matrix is unitarily diagonalizable. I will sketch the proof below.

Theorem: Let $A$ be an $n\times n$ complex matrix. Then $A$ is unitarily diagonalizable if and only if $A$ is normal.

Proof: I will sketch a proof for the backwards direction. Suppose that $A$ is normal. Then $AA^* = A^*A$. Let $A$ be triangularized by the unitary matrix $U$. We then have $T=U^*AU$. Now consider $$T^*T = U^*A^*AU = U^*AA^*U = TT^*$$ If you compare the diagonal entries of $T^*T$ and $TT^*$ you will see that the equality forces the off-diagonal entries to be zero.

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