Re: When are eigenvalues real for non-symmetric matrices?

On Tue, 8 Jan 2008 15:03:16 -0800 (PST), "Mariano Suárez-Alvarez"
<mariano.suarezalvarez@xxxxxxxxx> wrote:

On Jan 8, 8:19 pm, lilian.laco...@xxxxxxxxx wrote:
On 8 jan, 22:36, Yaroslav Bulatov <yarosla...@xxxxxxxxx> wrote:

Suppose A is diagonalizable nxn real matrix. When can we say that A
has no non-real eigenvalues, other than when it's symmetric?

I guess if A is diagonalisable then there exist a base B such that
A=B'*D*B where D is a diagonal matrix composed of the eignenvalues
(assume transposition of M is noted M'). And therefore you can show
that A'=(B'*D*B)'=(D*B)'*(B')'=B'*D'*B=B'*D*B=A (note that D'=D
because D is diagonal and hence symetric). So this is enough to show
that A is symetric.

Then if a real matrix has no non real eignenvalue, it is necessarily
symetric.

Like

(2 2)
(1 2),

for example.

I assume he meant to say "diagonalizable real matrix".

quasi
.

Relevant Pages