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Let $V$ be an inner product space generated over $\mathbb{C}$ and $B$ is a $n\times n$ normal complex matrix.

(1)I need to show that there exists a matrix $C$ such that $C^{2}=B$. I know that B is orthogonally diagonalizable by a theorem which was proved in class. Could I should that $C=BC^{-1}$?

(2) If the eigenvalues of $B$ are real, then $B$ is self-adjoint. Not sure where to start on this one.

  • 2
    Can you find a square root of a diagonal matrix with complex entries?2019-04-21

2 Answers 2

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Hint: Notice that if you have a matrix of the form $B=PDP^{-1}$ then $B^2 = (PDP^{-1})(PDP^{-1}) = PD^2P^{-1}$ Can you think of a square root for a diagonal matrix?

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    $A^* = (P^*DP)^* = P^*D^*P = P^*DP=A$2012-11-30
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Hint: $(ACA^{-1})^2=AC^2A^{-1}$