Write a Positive definite Matrix as $B^T B$

Question

I came across this problem:

Let $A$ (symmetric) be a positive definite matrix. Prove that there exist $n$ linearly independent vectors $x_1, x_2, ..., x_n$ such that $A_{ij} = x_i^T x_j$. (Hint: Use the spectral theorem and the fact that all eigenvalues of $A$ are greater than zero to find a matrix $B$ such that $A = B^T B$.)

I have no idea. Could someone please show me show to show this?

Answer 1

$$A = X D X^t = X \sqrt{D} \sqrt{D}^t X^t.$$