Showing something satisfies Inner Product - Involves Orthogonal Matrices

1. Oct 15, 2014

Circular_Block

1. The problem statement, all variables and given/known data

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.

2. Relevant equations
None

3. The attempt at a solution

I've shown that x · Dy is an inner product. I know that Z-1 is equal to ZT. I believe that will lead me somewhere. I'm just having trouble showing the property <x, x> ≥ 0. I also know that (ATx)⋅x = x ⋅ Ax.

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

Last edited: Oct 15, 2014
2. Oct 15, 2014

vela

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

3. Oct 15, 2014

Circular_Block

Doing that you'll end up with (ATx)⋅x right?

Last edited: Oct 15, 2014
4. Oct 15, 2014

Dick

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

5. Oct 15, 2014

RUber

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

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