I Spectral theorem for Hermitian matrices-- special cases

[nomadreid]

TL;DR Summary

I am not sure whether, for a finite Hermitian matrix M, a spectral decomposition M=PDP^-1 includes that (a) the P can be required to be unitary, and (b) whether the diagonal D can be required to be the classic upper-&-lower triangular kind

I have a proof in front of me that shows that for a normal matrix M, the spectral decomposition exists with
M=PDP^-1
where P is an invertible matrix and D a matrix that can be represented by the sum over the dimension of the matrix of the eigenvalues times the outer products of the corresponding basis vectors v_i
of an orthonormal basis, i.e.,
Σλ_i |v_i><v_i|

What the theorem does not say is whether, for Hermitian matrices (which are normal) M, one can require that the P be unitary. I seem to recall reading that it can but cannot find where I read that, and do not know how to prove it myself.

Also, the classic "diagonal matrix" is one with all the off-diagonal entries zero, but in this theorem, diagonal is just having the form above (with the summation), which need not have all zero off-diagonal entries. So I do not know whether the theorem would be valid if one required D to be a classic diagonal matrix, as I do not know how I would get from the above outer sum to a classic diagonal matrix.

{On less solid ground: my intuition (always a bad guide) suggests that one could rotate the orthonormal basis to a basis with (1,0,0...), (0,1,0...,)(0,0,1...)...., but that seems too reliant on a spatial intuition of perpendicularity. Nagging at me is also the idea that there are Hermitian operators that can't agree on a basis, so that it seems unlikely that they could all be reduced to the same basis even if they are separated by
P_P^-1. My intuition on this point is fishing around without any rigor, so anything shooting down this lead balloon is also welcome.]

Thanks for any help.

 

[PeroK]

http://scipp.ucsc.edu/~haber/ph116A/diag_11.pdf

 

[nomadreid]

Excellent. Thank you, PeroK.