[Math] Entries of a positive definite symmetric matrix as inner products

Question

For any $nxn$ positive definite symmetric matrix $A$ is it possible to write it's entries $a_{ij}$ as inner products of vectors $v_1,v_2,….,v_n$, that is $a_{ij}=\langle v_i,v_j\rangle$?
Is there a deterministic way to find $v_1,v_2,….,v_n$ for any given $A$?
I thought maybe Cholesky decomposition would be the best way to do it.
Have I missed something? Is there any easier way?

Thanks!

Answer 1

Probably Cholesky decomposition is one of the best ways, as once we have written $A=BB^T$, whose row vectors are $v_1,\dots,v_n$ we are done. Conversely, if the $v_j$ such that $a_{ij}=\langle v_i,v_j\rangle$ are given, then we get the Cholesky decomposition writing $B:=\pmatrix{v_1^T\\ v_2^T\\\vdots \\ v_n^T}$.