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[chuy52506]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

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

 

[gb7nash]

edited this post by mistake

 

[chuy52506]

so then on the right side would we have (AA^*)^-1?

 

[gb7nash]

So we have A*(A^-1)* = A^-1(A^-1)*

The right side is already in the form that we want. Look at the left side. We want the left side to eventually turn into the identity. If you ignore this problem for a second, given two random matrices P and Q, what's another way of writing Q*P* ? If you're still not sure, take a look at the wiki page:

http://en.wikipedia.org/wiki/Conjugate_transpose

Once you have this figured out, how can we rewrite A*(A^-1)* ?

 

[chuy52506]

ohh ok so we have (A^-1A)^* on the left side

 

[chuy52506]

which is just the identity

 

[gb7nash]

Correct

 

[chuy52506]

what about for the 2nd part?
would you start the same?

 

[lanedance]

well normal matrix is given by A*A=AA*

consider multiplying on the left by (A*)^-1A^-1 and on the right by A^-1 (A*)^-1