Finding the Inverse of a Normal Matrix

[chuy52506]

Homework Statement

Let A be a normal matrix in the complex field.

Homework Equations

Show that A^-1 is normal.

The Attempt at a Solution

I know that a normal matrix is AA^*=A^*A
what would i multiply this to start?

 

[Dick]

I would think multiplying by A^(-1) on the left would be a good start.

 

[chuy52506]

SO i would get A^*=A^-1A^*A
I think i would just get stuck wouldn't I?

 

[Dick]

SO i would get A^*=A^-1A^*A
I think i would just get stuck wouldn't I?

If that's all the farther you've gotten then you are stuck already. Why don't you just try something. You want to get to (A^(-1))*A^(-1)=A^(-1)(A^(-1))*. Just mess around until you get there. Remember you can always take '*' or '^(-1)' of both sides in addition to multiplying both sides by stuff.