Is it true that if $A$ is symmetric then any square root is symmetric? I can't prove this using basic symbolic computation, so what if we insist that $A$ is diagonalizable, or even positive definite?
2026-04-01 04:41:26.1775018486
A fact about symmetric matrices and square roots
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This is not true in general. For example: $$\pmatrix{0&-1\\1&0}^2 = \pmatrix{-1&0\\0&-1} .$$ Note that the matrix on the l.h.s. is diagonalizable (over $\Bbb C$).