Date: Nov 7, 2012 12:43 PM
Author: Kaba
Subject: Re: Uniqueness of Q
7.11.2012 18:01, Rupert wrote:

> On Nov 7, 4:39 pm, Kaba <k...@nowhere.com> wrote:

>> 7.11.2012 17:30, Kaba wrote:

>>> Hi,

>>

>>> Let Q, U, D, V in R^{n x n}, where Q^T Q = U^T U = V^T V = I, and D is

>>> non-negative diagonal. Consider the equation

>>

>>> Q^T UDV^T = VDU^T Q.

>>

>>> One solution to this equation is Q = UV^T.

>>

>>> Prove or disprove: this solution is unique.

>>

>> Disproved: Q = -UV^T is also a solution.

>>

>> Prove or disprove: UV^T and -UV^T are the only solutions.

>>

>> --http://kaba.hilvi.org

>

> Would it not be the case that any scalar multiple of UV^T is a

> solution?

>

> Also I think UD'V^T is a solution whenever D' is diagonal.

Oops, take that back. The solution Q = UD'V^T needs to orthogonal, so

all the elements in the diagonal of D' must be either +1 or -1. Anyway,

are _these_ all solutions?

--

http://kaba.hilvi.org