Matrix that has square root

A matrix A belonging to is a square root of B belonging to if =B. Show that every diagonalizable matrix in has a square root.

I know that a diagonal matrix has all zeros outside of the diagonal and the diagonal may or maynot have zeros. I want to say that these matrices must be similar as well.

I am not sure how to prove this. Any help would be great. Thanks!

Re: Matrix that has square root

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**page929** A matrix A belonging to

is a square root of B belonging to

if

=B. Show that every diagonalizable matrix in

has a square root.

I know that a diagonal matrix has all zeros outside of the diagonal and the diagonal may or maynot have zeros. I want to say that these matrices must be similar as well.

I am not sure how to prove this. Any help would be great. Thanks!

Here is the jist of it. If a matrix is diagonalizable then there exists a matrix P such that

where

Now define the matrix B as

Where

Now we have that