A positive definite matrix A can be express as a polynomial of $ A^2 $

Question

I have recently come across this question in my studies of matrix theory stating :

If A is a real symmetric positive definite matrix, we are asked to show it can be expressed as a polynomial of $ A^2 $

I thought about using the Cayley-Hamilton theorem via the characteristic polynomial, by I cannot guarantee only even powers of A in this matrix equation, so I am stuck and require help here. I thank all helpers.

Answer 1

Suppose $\lambda_i\geq 0$ are the distinct eigenvalues of $A$ and let $e_i(x)=\prod_{j\neq i}(x-\lambda_j^2)/\prod_{j\neq i}(\lambda_i^2-\lambda_j^2)$. Then $e_i(A^2)$ is a projection onto the eigenspace associated with $\lambda_i^2$ (of $A^2$), whence with $\lambda_i$ (of $A$). Then $$ A=\sum_i \lambda_i e_i(A^2)$$ If I am not mistaking one only need $A$ diagonalizable and the $\lambda_i^2$'s being distinct.

Answer 2

This has little to do with Cayley-Hamilton theorem. You only need to find a polynomial $p$ such that $p(\lambda_i^2)=\lambda_i$ over a set of distinct positive numbers $\lambda_1,\lambda_2,\ldots,\lambda_k$.