Re: Maximizing the trace
- From: lrudolph@xxxxxxxxx (Lee Rudolph)
- Date: 9 Apr 2005 20:28:27 -0400
garrettbaird2@xxxxxxxxxxx writes:
>Hello,
>Can anybody tell me what orthogonal matrices have maximum trace? It's
>the last part in a series of problems for me which I cannot figure out.
> We are given as a hint the Cayley parametrization, which says that any
>orthogonal matrix O that does not have -1 as an eigenvalue can be
>expressed as
>O = (I + A)(I - A)^-1
>where A is skew-symmetric. I tried using Lagrange multipliers but it
>got to be a big hairy mess. Any thoughts?
Yeah, my thought is that that hint is a mean trick, or I
totally misunderstand something. Each column of an orthogonal
matrix is a vector of norm 1, right? So each entry in each
column (and therefore each entry in the matrix) is between -1
and 1. So the identity matrix has trace n (the size of the
matrix), and every other orthogonal matrix has trace strictly
less than n.
Lee Rudolph
.
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