Let A be an n × n real diagonalizable matrix. Show that A + αIn is also real diagonalizable.

Question

Let A be an n × n real diagonalizable matrix. Show that A + αIn is also real diagonalizable.

I am having trouble figuring out where to start. I know that if I show that A + αIn has n distinct real eigenvalues, then it is diagonalizable, but I don't know how to show it.

Answer 1

There is an easier way : assume that $A$ is diagonalizable, i.e. $A=PDP^{-1}$ with $D$ diagonal matrix and $P$ invertible matrix. Then write $$A+\alpha I_n=PDP^{-1}+\alpha PP^{-1}=P(D+\alpha I_n)P^{-1}$$ and as $D+\alpha I_n$ is a diagonal matrix you have your result.