A short one on symmetric matrices

[Päällikkö]

This isn't really homework, but close enough. I suppose this is quite simple, but my head's all tangled up for today. Anyways,
Given the real symmetric matrix
L^TL = UDU^T, find L.

I suppose L = +- D^1/2U^T, and it's clear this choice of L satisfies the given equation.

But can it be proven that the above L actually follows from the given equation? i.e.
L^TL = UDU^T = (D^1/2U^T)^T(D^1/2U^T) <=> L = +- D^1/2U^T?
Am I making any sense?

 

[Dick]

So you are asking if L is unique? No. Take L and U to be different unitary matrices and D=1. Then T(L).L=1=U.D.T(U) (T=transpose). But it certainly isn't necessarily true that L=+/-T(U).