Re: Tell whether a matrix is full rank using a calculator
- From: Chip Eastham <hardmath@xxxxxxxxx>
- Date: Tue, 14 Aug 2007 17:14:46 -0000
On Aug 14, 12:58 pm, isabelle...@xxxxxxxxxxx wrote:
Hello all,
I would like to be able to tell if a (say 5*5) matrix is full rank by
using my graphic calculator.
My calculator does not have a built in function that computes the
rank. Is there another way to know whether a matrix is full rank? Say
by looking at the eigenvalues (my calculator will compute the
eigenvalues), or some other method....
I greatly appreciate your input
If arithmetic were exact, then the absence of zero
among the eigenvalues would be equivalent to full
rank.
To deal with inexact arithmetic/rounding errors,
one needs to set some sort of threshold for the
notion of eigenvalues reported as so near to zero
that the matrix should be considered singular
(not full rank). Ordinarily we normalize by
using the ratio of the smallest eigenvalue to
the largest one in absolute value, and asking
whether this ratio is below some small multiple
of epsilon (machine precision).
I suppose the choice of what multiple to use
depends on how conservative you wish to be in
calling a matrix singular when in fact it is
only nearly singular.
In a richer setting (than a graphing calculator)
one might use a QR factorization to squeeze out
the best rounding error behavior.
regards, chip
.
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