How can I show from $AA'=PP'=C$ that $A$ is an orthonormal transformation of $P$?

Question

Problem details

• $P$ is the lower triangular Cholesky factor of the positive semidefinite matrix $C$ ($PP'=C$)
• $A$ is any matrix such that $AA'=C$

How can I show that $A$ is an orthonormal transformation of $P$?

My thought process so far

It not necessarily the case that $AA'=I$, which was my first attempted route for solving the problem

I then discovered that if $A$ is an orthogonal transformation of $P$ then $\langle AP, AP \rangle = \langle P, P \rangle$ but I do not know how to apply it.

Any help is appreciated

Answer 1

Let $P = S_1O_1$ and $A = S_2O_2$ be polar decompositions of $P$ and $A$ respectively, where $S_i, i = 1, 2$ are symmetric positive semidefinite matrices, and $O_i, i = 1, 2$ are orthogonal matrices. Then $C = PP' = AA'$ implies $C = S_1S_1' = S_2S_2'$. Or equivalently $C = S_1^2 = S_2^2$. Since the square root of a positive semidefinite matrix is unique, it follows that $S_1 = S_2$, i.e., $PO_1' = AO_2'$. Therefore, $A = PO_1'O_2$. Since the product of orthogonal matrices are still orthogonal, this completes the proof.