Re: eigenvector for a 3-by-3 symmetric real matrix


"agou" <agou.win@xxxxxxxxx> wrote in message
news:1175746181.292952.303120@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
hi, there

i wanna calculate the eigenvector to the smallest eigenvalue for a 3-
by-3 symmetric positive definite real matrix. i know it might sound
silly. however, i wanna put this stuff running on hardware. therefore,
analytic result would be best. however, it's not trivial. therefore,
maybe an iterative but fast converging algorithm will be preferred. QR
looks like too complicate either. does anyone have any better idea or
book i could refer to?

thanks a lot!

regards
zhaoyi

For such a small example you could simply find ythe characteristic equation
and solve the resulting cubic to get the eigenvalues and then from there
find the eigenvectors.

If you have an idea what the eigenvalue roughly is then you can use Inverse
Iteration or Rayleigh Quotient Iteration to find more precisely the
eigenvalue, and also at the same time the corresponding eigenvector.

If you're not going to use the QR algorithm (which isnt particularly
difficult to implement by the way), then there are other methods for special
case matrices but I'm sure they are more for very large matrices rather than
small ones like a 3 x 3. I'm sure Peter Spelluci can tell you far more
about this than I can.

Jeremy Watts




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