Thread Subject: Question about Matlab function eig

The eigen decomposition funcion eig in Matlab is supposed
to return orthonormal vectors. Eg. given matrix A, we have
V'AV=D. ideally, if the eigen vectors are authonormal to
each other, we should have V'V=I, however, if you run the
following command:

clc, A=rand(3,3), [V,D] = eig(A), V'*V,

You'll find this is not always true. The V'*V is not
identity matrix. Also, for an asymmetric matrix A, we
expect to get complex eigen value and eigen vectors. But
if you run above command, you'll see it is not always true
either.

Can anyone explain this?

"Tian Lan" <lantian@mathworks.com> wrote in message
<fkc4he$g1m$1@fred.mathworks.com>...
> The eigen decomposition funcion eig in Matlab is supposed
> to return orthonormal vectors. Eg. given matrix A, we have
> V'AV=D.

That is not true. Your should reread the doc of eig (and
probably algebra course).

Bruno

"Tian Lan" <lantian@mathworks.com> wrote in message <fkc4he$g1m
$1@fred.mathworks.com>...
> The eigen decomposition funcion eig in Matlab is supposed
> to return orthonormal vectors. Eg. given matrix A, we have
> V'AV=D. ideally, if the eigen vectors are authonormal to
> each other, we should have V'V=I, however, if you run the
> following command:
>
> clc, A=rand(3,3), [V,D] = eig(A), V'*V,
>
> You'll find this is not always true. The V'*V is not
> identity matrix. Also, for an asymmetric matrix A, we
> expect to get complex eigen value and eigen vectors. But
> if you run above command, you'll see it is not always true
> either.
>
> Can anyone explain this?
--------
  Eigenvectors are not always mutually orthogonal. Only when A is Hermitian
symmetric is that necessarily so. Also it isn't true that asymmetric matrices
give complex eigenvectors and eigenvalues, as you have found out. There
will be complex eigenvalues whenever there are any complex roots to the
characteristic equation, but it is easy for that equation to have only real roots
stemming from an asymmetric matrix.

Roger Stafford