Any good books to practice on Endomorphisms as related to Diagonalization, Cayley-Hamilton, etc.?

Question

Well, I am looking for books (graduate level) that covers linear maps (endomorphisms, to be specific) with emphasis on topics related to numerical linear algebra, like: diagonilzation, triangularization, Gram-Schmidt process, etc.

I looked up few books but they either too abstract for me (and for the course treatment), or don't have any exercises or even solution manual.

Here is an example of some exercises the professor asked us to look at and expect: https://i.stack.imgur.com/hBSoH.png

Another example is from a graduate algebra course at Kent State university: http://www.personal.kent.edu/~akasturi/tom/hw6.pdf

Any help would be appreciated!

Answer 1

Every single example that you've given has either to do with the Cayley Hamilton theorem or Jordan canonical form. In fact, I can sum it up in 2 lines:

• Use the Cayley-Hamilton theorem
• If $p(x)$ is a polynomial such that $p(A) = 0$, then the minimal polynomial of $A$ divides $p$.

A suitable reference for any of these questions is Matrix Analysis by Horn and Johnson. In particular, chapter 3 (sections 0-3) tell you everything you need to know for these questions.

The Cayley-Hamilton theorem is proved in chapter 2, using the Schur decomposition.