Spectral Mapping Theorem
Let 
be a square matrix of order 
, and 
a polynomial, then show 
, where 
is the spectrum of 
– the set of eigenvalues. This result is known as the spectral mapping theorem, and can be written more concisely as
One inclusion is trivial (
), the other is trickier.
Now use the spectral mapping theorem to characterize the spectrum of an idempotent matrix (i.e., 
).
Here’s a harder one: can we have matrices 
such that 
?
Possibly relevant posts:
- The Gelfand representation theorem (7/4/2008)
- Companion matrix (10/12/2005)
- Two ways of doing cool stuff with matrices (2/27/2007)
respected sir, i am having a doubt. the spectrum of zero operator is what?
Comment by G.devipriya — 12/14/2006 @ 7:25 pm
the empty set.
Comment by Anonymous — 3/28/2008 @ 3:14 pm