Is a convex combination of positive semi-definite matrices itself positive semi-definite?

Question

Suppose I have $\Sigma_1, \dots, \Sigma_n$ all positive semi-definite, and $\lambda_1, \dots, \lambda_n$ such that $\lambda_i \geqslant 0$ and $\lambda_1 + \dots + \lambda_n = 1$. Is it true that $\Sigma = \lambda_1\Sigma_1 + \lambda_n\Sigma_n$ is positive semi-definite?

Answer 1

A matrix $A$ is positive semidefinite if and only if it is symmetric and $x^TA x\ge 0$ for all $x\in\Bbb R^n$. Since $x^T(\sum_j \lambda_jA_j)x=\sum_j\lambda_jx^TA_jx$, the inequality passes to linear combinations by non-negative coefficients (and of course so does symmetry).