Functions of matrices are typically defined by their Taylor series expansions. For example,

(a) Find exp (M), if

(b) Show that if M is diagonalizable, then

Comment: This is actually true even if M is not diagonalizable, but it’s harder to prove in the general case.
(c) Show that if the matrices M and N commute, then

Prove (with the simplest counterexample you can think up) that Equation A.101 is not true, in general, for non-commuting matrices.
(d) If H is hermitian, show that e^iH is unitary.

Transcribed Image Text:

eMI+M+M² + -M³ 3! +. (A.99)