Show a matrix is normal - check my proof

Question

Short easy question, I just want someone to double check what I did.

We are given that $T$ is an invertible, normal matrix. We are asked to show that $T^{-1}$ is also normal, and find it's unitary diagonlization.

What I did:

$T$ is normal if and only if there is a unitary matrix $U$ and diagonal matrix $D$ such that $T=UDU^{-1}$

if $T=UDU^{-1}$ then $T^{-1}=(UDU^{-1})^{-1} = U^{-1}D^{-1}U$

$U$ and $U^{-1}$ are still the same (unitary) and $D^{-1}$ is still diagonal. So $T^{-1}$ is unitary diagonlizable, so it is normal, and as stated above, the unitary decomposition is $T^{-1}=U^{-1}D^{-1}U$

Answer 1

Your answer is almost correct (though uses a nontrivial theorem about normal matrices). Two missing/incorrect points:

• If $T=UDU^{-1}$ then $T^{-1}=(UDU^{-1})^{-1}=UD^{-1}U^{-1}$ (taking inverse changes order of multiplication).
• As imj noted, we should also argue for the invertibility of $D$ before (or after) we write it down, but it is easy now: as the above equation implies, the matrix $U^{-1}T^{-1}U$ will be an inverse for $D=U^{-1}TU$.

Answer 2

You can go also directly from the definition.

$$TT^*=T^*T$$

Invert both sides.

$$(T^*)^{-1}T^{-1}=T^{-1}(T^{*})^{-1}$$

Use that $(T^{*})^{-1}=(T^{-1})^*$, which also folows directly from the definition.