Prove that Let A be a definite positive matrix, such that for every positive integer k, there exists a symmetric matrix B such that A=B^k

Question

I need some help, I need to prove the following exercise:

Let A be a definite positive matrix, such that for every positive integer k, there exists a symmetric matrix B such that A=B^k

I haven't thought how to solve it, can anyone give me a hint? Thank you!

Answer 1

If $A$ is positive definite, its eigenvalues are all strictly positive, and an eigendecomposition exists, $$ A = PDP^{-1}. $$ where $D$ is a diagonal matrix. Then take $$ B = P D^{1/k} P^{-1} $$ so that $A = B^k$.