Can you help find a $2\times 2$ matrix with eigenvalues $1,-1$ that is not a normal matrix?
I really tried to find one but the matrix I found is also normal!!$A$ hermitian and $B$ unitary matrix and AB=BA I need to show $AB$ is normal matrix.
Well I know that both $A,B$ are normal and that $A=A^*$ and $B=B^{-1}$
Then $(AB)(AB)^* = (AB)(A^*B^*)$ but now Im not sure what is ok to do for showing $AB$ is normal...
Normal and unitary Matrices
3
$\begingroup$
linear-algebra
matrices
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0$(AB)^{\ast}=B^{\ast}A^{\ast}$. – 2012-12-21
1 Answers
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Hint for (1): $A$ has the same eigenvalues as $S A S^{-1}$ for any invertible matrix $S$. Try an $S$ that is not normal.
(2): No, $B^* = B^{-1}$ and $(AB)^* = B^* A^*$. But the statement is not true: if $A$ is hermitian and $B$ is unitary, $AB$ is not normal unless $A^2$ commutes with $B$.
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0strange that is what I need to prove... – 2012-12-21
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0Oh I forgot something I edited my question now... – 2012-12-21
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0Are you sure there isn't some other assumption, e.g. that $A$ has eigenvalues $1$ and $-1$? – 2012-12-21
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1Oh, now it's easy. In fact the product of two normal matrices that commute is normal. – 2012-12-21
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0Yes thank you for Question2 but still cant understand 1 – 2012-12-21
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0You might try a lower (or upper) triangular matrix. – 2012-12-21
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0oh thats was so easy... thank you! – 2012-12-21