If M is diagonalizable, find a matrix N^3 = M

1. May 15, 2014

victoranderson

I have a question about diagonalizable matrix

So now I have showed M is a diagonalizable and I am asked to find a matrix N^3=M

Obviously if M=PDP^(-1)
then N=PD^(1/3)P^(-1) but I am wondering how to show it formally.

2. May 15, 2014

D H

Staff Emeritus
Given that $N=PD^{1/3}P^{-1}$, what is $N^3=NNN$?

3. May 15, 2014

haruspex

Strictly speaking, also need to show that this generates all solutions of the original equation.

4. May 16, 2014

D H

Staff Emeritus
I'm not sure what you mean by that. Per the title of the thread, victoranderson has to find a matrix N such that N^3=M, not all matrices N such that N^3=M.

What he does need to show is that his D^(1/3) always exists and that (D^(1/3))^3=D.

5. May 16, 2014

True.