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nomadreid

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- 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

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

of an orthonormal basis, i.e.,

Σλ

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

Thanks for any help.

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.