Square Root of a Diagonalizable Matrix

Question

I'm trying to find a reference that specifically covers taking the square root of a matrix that is diagonalizable.

I'm already combed through $\textit{Functions of Matrices: Theory and Computation}$ by Higham, but couldn't seem to locate it.

Answer 1

If the matrix is positive semi-definite, the usual version of this is to write $$ M = S^{-1} D S $$ for some matrix $S$ and diagonal matrix $D$. Let $E$ be the diagonal matrix whose diagonal entries are the square roots of the diagonal entries of $D$. Then $$ (S^{-1} E S)(S^{-1} E S) = S^{-1} EE S = S^{-1} D S = M $$ so $S^{-1} E S$ is a good thing to call "a square root of $M$".