# Good resources on matrix equations

I’m searching for an online resource, preferably free.
I refer to equations such as $$A^3 -A+ I=0$$ Or even something like: $${A^t}^2+A^2=I$$ I know that normal factorization methods don’t work as matrices aren’t a field.
I’m not looking for a general linear algebra resource, I’m looking for a resource specifically about matrix equations.

• FYI, you can do this using the Jordan form. Also, try googling "minimal polynomial". – Adam Jul 11 '20 at 22:42

Take a look at the Naval Academy course on Matrix Theory. It emphasizes matrix manipulation more than proofs shmoofs. And is not a general LA class.

https://www.usna.edu/Users/math/hottovy/Teaching/SM261.php

There are some worksheets and notes you could look at also on the SM261 page.

Here is a link to Hefferon's LA text (used for the 261 class), which is progressive and FREE and emphasizes matrix manipulation at the start.

http://joshua.smcvt.edu/linearalgebra/

Other than that, I heartily recommend the chapter on matrices in Kreyszig's Advanced Engineering Mathematics. It is very self-standing and gives you exactly what you want (learning to rock the matrices around the clock, not abstraction for the proof lovers). It's not free, but you can get used copies (edition not important, I liked the 5th). About 50% of the problems had answers in the back, so helpful for drill. Also, you can just look at in the library (maybe even zox the chapter).

But as I commented above, you can do it for yourself if you think about the fact that $$A$$ and $$P^{-1}AP$$ will always be solutions to the same matrix polynomials. Then solve the equation for the case where $$A$$ is in Jordan form. All other solutions will be conjugates of these type of solutions. (There will often be several Jordan forms which solve the equation, depending on multiplicity of roots, etc.)