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$A$ is $n\times n$ matrix over $\mathbb C$. must exist that:

  1. $A^*A$ is diagonalizable over C
  2. $AA^*$ is unitary matrix
  3. if $A$ is not diagonalizable over $\mathbb C$ so $AA^*$ is not diagonalizable over $\mathbb C$
  4. $i+1$ is not eigenvalue of $A$

I know the answer is 1+4 but I really dont understand why!

  • sorry for bad english
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1 Answer

up vote 4 down vote accepted
  1. It's true, as any Hermitian matrix is diagonalizable over $\Bbb C$.
  2. It's not necessarily true, for example if $A=2I$, $AA^*=4I$ is not unitary.
  3. $AA^*$ is always diagonalizable over $\Bbb C$ (as a Hermitian matrix), independently of the fact that $A$ is diagonalizable or not.
  4. May happen ($A=I$ for example) or not (if $A=(1+i)I$).
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but in 1 how do I know A is Hermitian?? –  Mary Aug 3 '12 at 17:52
    
$A$ may be not Hermitian, but $A^*A$ is Hermitian (it follows from properties of the adjoint). –  Davide Giraudo Aug 3 '12 at 17:59
    
thank you very much! –  Mary Aug 3 '12 at 18:16
    
You are welcome! –  Davide Giraudo Aug 3 '12 at 18:16
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