TITLE: In a finite dimensional vector space a positive operator can have infinitely many square roots
QUESTION [5 upvotes]: "In a finite dimensional vector space a identity operator (which is positive) can have infinitely many square roots". I see only $2 ^ {C(n,2)}$ when dimension is $n (>1)$ by swapping two basis elements. Where am I wrong ?

REPLY [4 votes]: To elaborate on your example, $Y=\pmatrix{0&1\\ 1&0}$ is a square root of $I$. Therefore, $X=PYP^{-1}$ is also a square root of $I$ for every invertible $P$. One can obtain infinitely many such $X$s by using different $P$s. And basically, all square roots of $I$ are obtained in this manner, i.e. they are similar to a diagonal matrix whose diagonal entries (or eigenvalues) are $\pm1$.