Re: The dimension of fixed space of a matrix

On Aug 26, 1:57 am, Robert Israel
<isr...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
Gvnaena Pura <tianran.c...@xxxxxxxxx> writes:
I'm wondering if there is an easy way for knowing the dimension of the
fixed subspace of a matrix, when it act on a vector space. Here, by
the fixed subspace, what I mean is the subspace W such that Mx = x for
all x in W where M is the matrix in question. It is also the kernel
of the map (I-M), where I is the identity. (maybe there is a formal
for this, but I don't know). Thanks in advance.

So if M is n x n, you want n - rank(M-I). To actually find it numerically,
use Gaussian elimination or singular value decomposition.

Thanks. But I'm interested in knowning the dimension a priori. Maybe
in general one cannot answer this question, but what about in some
special cases? For instance, can we find dim(Ker(I-M)) for M an
orthogonal/unitary matrix? Thank in advance.

.

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