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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?


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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 $B:=\pmatrix{v_1^T\\ v_2^T\\\vdots \\ v_n^T}$.

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"every positive semidefinite matrix is the Gramian matrix for some set of vectors.", see this.

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