Can a positive definite matrix $S$ always be expressed in the form $S=A^TA$?

Question

Can a positive definite matrix $S$ always be expressed in the form $S=A^TA$?

If so, is there a simple proof of this? I ask the question because another proof in the book Intro to Linear Algebra by Strang uses this fact.

Answer 1

Assuming the matrix is real and symmetric, then yes it can. One way of doing it is known as the Cholesky decomposition. In fact, here we may even assume that $A$ is upper triangular, with positive entries on the diagonal.

(Note that the Wikipedia article mostly assumes that $S$ has complex entries, and that it is Hermitian rather than symmetric. However, in the "Statement" paragraph, the article mentions that the decomposition works for real matrices too, and in that case, $A$ will also have real entries. The original result by Cholesky was also for real matrices, and it was only later generalized to the complex case. Also, Wikipedia transposes the second factor, not the first, so their matrix is lower triangular instead.)