a.) Prove that every invertible matrix A has a polar decomposition, written $A = QB$, into the product of an orthogonal matrix $Q$ and a positive definite matrix $B>0$. Show that if $detA>0$, then $Q$ is a proper orthogonal matrix.
c.) Prove that every positive definite matrix $K$ has a unique positive definite square root.
How will I be able to prove these?
For part a - I know I must use Gram matrix $K = A^TA$ in order to prove it.
For part c - I do not know what to do.