Eigenvalue Factorization and Matrix Substitution

In my literature reviews I found a few things that I can't quite understand.
1. The problem statement, all variables and given/known data

I have the following equation:

I'm told that by using the eigenvalue factorization:

, I can change the first equation to:

2. The attempt at a solution

I tried changing Equation 2 to just be (A^T)A and then subbing into the first equation, but I can't quite do anything with those inverses.

Also, what does the exponent of '-2' mean in the context of a 4x4 matrix? Lastly, what is matrix U?

Thank you!
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Recognitions:
Homework Help
 Quote by the_dialogue I tried changing Equation 2 to just be (A^T)A and then subbing into the first equation, but I can't quite do anything with those inverses.
I think it's probably easiest to start from $\textbf{p}^{T}(\mathbf{\Lambda}+\lambda\textbf{I})^{-2}\textbf{q}=0$ and work your way backwards instead.

 Also, what does the exponent of '-2' mean in the context of a 4x4 matrix? Lastly, what is matrix U?
$$\textbf{C}^{-2}\equiv\textbf{C}^{-1}\textbf{C}^{-1}$$

You simply square the inverse of the matrix.
 I'll give it a try gabbagabbahey. Thanks. Any idea what the matrix "U" is?

Recognitions:
Homework Help

Eigenvalue Factorization and Matrix Substitution

 Quote by the_dialogue Any idea what the matrix "U" is?
It's the invertible matrix which relates the matrix $\textbf{A}^{T}\textbf{A}\mathbf{\Sigma}$ to the diagonal matrix $\mathbf{\Lambda}$ via a similarity transform. Its columns will be the eigenvectors of $\textbf{A}^{T}\textbf{A}\mathbf{\Sigma}$.

See http://en.wikipedia.org/wiki/Diagonalizable_matrix for a refresher on matrix diagonalization.
 Yes I recall now. Thanks!