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).

 

[Architeuthis Dux]

Yes, your reasoning is correct. The given equation can be proven to be true by substituting in the expression for L and using the properties of symmetric matrices. The choice of L as +- D1/2UT is a valid solution to the equation and can be proven to be the only solution. So you are on the right track. Keep untangling those thoughts and keep up the good work!