Question

Let \(\{v_1,v_2,...,v_n\}\) be a set of vectors in an inner product space and let \(A\) be the \(n \times n\) matrix with \(A_{i,j}\) equal to the inner product of \(v_i\) and \(v_j\). Show that \(A\) is a symmetric positive semidefinite matrix.

Answer 1

Symmetric part is trivial but it may spark an idea...

Answer 2

So I want to show \(x^TAx\ge 0 \ \forall x\). So \(\sum_{i,j=1}^nA_{i,j}x_ix_j\ge 0 \) and I just answered my own question. boo sorry.

Answer 3

sum of squares is never negative
Easy peasy!

Answer 4

funny how just writing something out will often solve itself.