For each of the following statements, prove that it is true or give an example to show it is false. throughout, a e gm x r n unless otherwise indicated, and "ew" stands for eigenvalue. (this comes from the german "eigenwert." the corresponding abbreviation for eigenvector is "ev," from "eigenvektor." )

( a ) i f x i s a n e w o f a a n d p ~ c , t h e n x - p i s a n e w o f a - p i .
(b) if a is real and x is an ew of a, then so is -a.
(c) if a is real and x is an ew of a, then so is 5; .
(d) if x is an ew of a and a is nonsingular, then a-i is an ew of a-'.
(e) if all the ew's of a are zero, then a = 0.
(f) if a is hermitian and x is an ew of a, then ixi is a singular value of a.
(g) if a is diagonalizable and all its ew's are equal, then a is diagonal