How can I prove the inverse of a unitary matrix is unitary?

[chuy52506]

A is a matrix in the complex field
Suppose A is unitary show that A^-1 is unitary.

Suppose A is normal and invertible, show A^-1 is normal.

Can i prove the first one just by:
AA^T=I
then A^T=A^-1

Then
I=A^-1(A^T)^-1
So,
I=A^-1(A^-1)^T


I have no idea in how to start the second one? please help

 

[Simon_Tyler]

I assume that when you write A^T you mean the Hermitian conjugate of A. For the sake of clarity, I'll use A^H:=(A^*)^t to mean the Hermitian conjugate.

Then yeah, your first proof is ok provided you're happy with the fact that
(A^H)^-1 = (A^-1)^H ----- (**)
which isn't that hard to prove.

Here's a slightly different proof:
A unitary <==> A^HA = I , ( where I = unit matrix)
take inverse of both sides (can do because it's assumed invertible)
A^-1(A^H)^-1 = I
multiply on left by A and right by A^-1
A A^-1(A^H)^-1A^-1 = (A^H)^-1A^-1 = I
use the result (**) and you're done.

Basically the same proof works for normal matrices.
A normal <==> A^H A = A A^H
Take the inverse
A^-1 (A^H)^-1 = (A^H)^-1 A^-1
Use the result (**) and you see that A^-1 must be normal
A^-1 (A^-1)^H = (A^-1)^H A^-1 .