Re: Matrix Math Question

Rune Allnor <allnor@xxxxxxxxxxxx> writes:

On 28 Jul, 19:41, Tim Wescott <t...@xxxxxxxxxxxxxxxx> wrote:
I just discovered that not all matrices with all-positive eigenvalues are
positive definite (dumb, yes, and I probably used to know it, yes, but
none the less...).

I'm doing some testing, and I need to reliably generate otherwise-random
matrices that are (a) positive definite and (b) have predetermined
eigenvalues.

Anyone got any suggestions?  I know how to do the part where I generate a
vector or matrix of random numbers, I'm just not sure where to go from
there...

The useful property of a positive definite matrix X
is the eigenvalue decomposition:

X = V'DV

where V is an orhogonal matrix,

V'V = I

and D is diagonal, containing the eigenvalues.

You already know the eigenvalues, so D is known.
The question, then, is how to come up with a
useful V of dimension N x N.

What I would do:

1) Generate a random vector v of size N x 1
2) Normalize the vector v to length 1
3) Use a Gram-Schmidt expansion around v
to find a complete N x N orthogonal
basis V.
4) Once you have that basis V, you are done.

In matlab the function QR performs step 3
when fed a vector.

Rune,

My LA book [meyer] tells me that the QR factoriation of an NxM matrix A produces
a matrix Q of dimension NxM and an upper-triangular matrix R of
dimension MxM, so that A = QR.

So by that definition, the QR factorization of the Nx1 vector A would be
an Nx1 vector Q and a 1x1 scalar R.

This also agrees with the notion of Gram-Schmidt orthogonalization
for the case of a set of 1 Nx1 vector {A}. The space spanned by
that vector is just range(A), and you'd expect that the dimension
of such as space to be one. Since the columns of Q form a basis
for range(A), you'd thus expect Q to have just one column.

But, as you've intimated, that isn't what Matlab (or Octave) returns. It
returns an NxN matrix Q and Nx1 vector R. What exactly is going on,
then?

--Randy

@BOOK{meyer,
title = "{Matrix Analysis and Applied Linear Algebra}",
author = "{Carl~D.~Meyer}",
publisher = "Society for Industrial and Applied Mathematics",
year = "2000"}

--
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Digital Signal Labs % and kiss her interface,
mailto://yates@xxxxxxxx % til then, I'll leave her alone."
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