Showing something satisfies Inner Product - Involves Orthogonal Matrices

[Circular_Block]

Homework Statement

[/B]
Let Z be any 3×3 orthogonal matrix and let A = Z^-1DZ where D is a diagonal matrix with positive integers along its diagonal.
Show that the product <x, y> _A = x · Ay is an inner product for R³.

Homework Equations

None

The Attempt at a Solution

I've shown that x · Dy is an inner product. I know that Z^-1 is equal to Z^T. I believe that will lead me somewhere. I'm just having trouble showing the property <x, x> ≥ 0. I also know that (A^Tx)⋅x = x ⋅ Ax.

Just missing one step. I don't know what it is.

 

[vela]

Hint: $(Zx)^T = x^T Z^T$

 

[Circular_Block]

Doing that you'll end up with (A^Tx)⋅x right?

 

[Dick]

Doing that you'll end up with (A^Tx)⋅x right?

x.y=x^Ty. That turns a dot product into a matrix product. Add that to the list of clues.

 

[RUber]

Don't neglect that D is all positive. So $Z^T D Z$ should also be non-negative, right?