Normalizing a 4x4 matrix with unknown functions as elements

[fruity_water]

Hi All!

This is my first time putting up my own thread on MF. I can usually find what I'm looking for, but this time: no go.

As the title says, I'm trying to find the normalization constant of this 4x4 matrix (g and f are functions):
(1-g^2) 0 0 0
0 (1+f^2) (-g^2-f^2) 0
0 (-g^2-f^2) (1+f^2) 0
0 0 0 (1-g^2)

It's a matrix that's in a research paper which gives the normalization constant as: N=4-2g^2+2f^2.

1]I've been looking up online and found that N can be found with: N=\sqrt{\sum{X^2}} where X represents the elements of the matrix.

2]I also found somewhere which said that I need to find the determinant.

I'm not sure who's right, but I'm not getting what's in the paper.
For method [1] I'm getting as far as: N^2 = 4(1+f^4+f^2g^2+f^2) and got stuck trying to find the square root (it's been a while since I've done multinomial theorem). So I backtracked to see if their N^2 matches my N^2. But their N^2=16+4g^4+4f^4+16g^2-8g^2f^2+16f^2.
and method [2] is giving me something so long, with so many variables of (g^2), (f^2), (g^4), (f^4),(g^2f^4) (and it keeps going for about 3tysomething variables) that I've given up.

So I'm wrong all over the place.

Can someone help me out?

fruity_water confused

 

[fresh_42]

I'm not quite sure which normalization you are looking for, there are various normal forms. An easy first step is to conjugate the matrix with the matrix $\begin{bmatrix}1&0&0&0\\0&1&1&0\\0&0&1&0\\0&0&0&1\end{bmatrix}$ and get
$$
\begin{bmatrix}1&0&0&0\\0&1&1&0\\0&0&1&0\\0&0&0&1\end{bmatrix}\begin{bmatrix}1-g^2&0&0&0\\0&1+f^2&-g^2-f^2&0\\0&-g^2-f^2&1+f^2&0\\0&0&0&1-g^2\end{bmatrix}\begin{bmatrix}1&0&0&0\\0&1&-1&0\\0&0&1&0\\0&0&0&1\end{bmatrix}=\begin{bmatrix}1-g^2&0&0&0\\0&1-g^2&0&0\\0&-g^2-f^2&1+g^2+2f^2&0\\0&0&0&1-g^2\\&&&\end{bmatrix}
$$
which is a lower triangular matrix. It allows you to directly see the eigenvalues.