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If a matrix A is defined by $(A)_{ij}=(X_i)^*X_j$, show that eigenvalues of A are non negative.

I was able to show that A is hermitian and an eigenvalue $\lambda$ can be written for an eigenvector Y as: $\lambda=(Y)^*(AY)/((Y)^*Y)$.

However I am unable to show that this is non negative.

Thanks for the help.

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Hint: Note that $$ Y^*AY = \left(\sum_{i=1}^n Y_i X_i\right)^*\left(\sum_{i=1}^n Y_i X_i\right) $$

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Let's $x_i = (X_i)^*$, then $A = xx^*$. Now if $y$ is eigenvector of $A$ we have $$ Ay = xx^*y = (x^*y)x = \lambda y, $$ so $y = \alpha x$ for some number $\alpha$, which implies $\lambda = x^*x = \sum x_i^*x_i = \sum |x_i|^2 > 0$.

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