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

[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.

 

[D H]

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

 

[haruspex]

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

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

 

[D H]

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

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.

 

[haruspex]

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.

True.