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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.

Pubbie
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  • Thanks for the links. The first link will definitely be usable, although I was hoping I could find it in a textbook somewhere. – Pubbie Nov 10 '14 at 23:45

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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 $M$".

John Hughes
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  • Right, so essentially it is just first finding $D$ such that $A=SDS^{-1}$, then we have that $\sqrt{A}=S\sqrt{D}S^{-1}$, but i'm having trouble finding a credible, academic source for this. – Pubbie Nov 10 '14 at 23:42
  • Well...I've got a Ph.D. in mathematics, held academic mathematics positions at Bryn Mawr and Brown, and now am a professor of computer science. :) For a more reputable source, Strang, Linear Algebra and its applications, 1998 edition, page 334, 2nd paragraph, tosses this idea off in passing. On page 337, exercise 6.2.7 gives more detail. – John Hughes Nov 11 '14 at 00:45
  • Nice! Haha, I wish I knew you IRL so I could just get a quote or something! :P But i'll take a look. Thanks for the tip. – Pubbie Nov 11 '14 at 04:14