For some integers $n$ and $K$, nonnegative scalars $(\lambda_a)_{1 \leq a \leq K}$ and vectors $(y_a)_{1 \leq a \leq K} \in \mathbb{R}^n$, consider the matrix $$A:=\sum_{a=1}^{K}\lambda_ay_ay_a^T.$$ The matrix A is clearly positive semi-definite. What are some conditions on $\lambda$ and $y$ for $A$ to actually be positive definite (i.e. when is it invertible) ?
2026-03-27 15:19:14.1774624754
necessary and sufficient conditions for inversibility of a specific positive semi-definite matrix
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