Hinted from Math 2270 – Lecture 33 : Positive Definite Matrices, by Dylan Zwick, foot note of page 4.

If is symmetric then is always diagonalizable: , . Set (), we have:

Hinted from Introduction to Linear Algebra – Gilbert Strang [WORKING AREA]

We have:

Now consider the expression , with is the entry at position of the matrix . Now also consider the expression: . We have because is a symmetric matrix. So yes, can also be interpreted like this:

is always a pivot. For some , we can group an expression like this:

So:

Note that because is symmetric, and yields the pivot at position . are in fact multipliers from elimination.