characteristic summation of a positive definite matrix and a covariance matrix of a non-square matrix with positive elements

Question

I have tow matrices $A_{n \times n}$ and $B_{n \times m}$ which $m<n$. $A_{n \times n}$ is a diagonal positive definite and $B_{n \times m}$ is a non-square matrix with positive or zero elements.
I want to find that If matrix $C_{n \times n}$ is semi-positive definite or not:

$C_n = A_{n \times n} + B_{n \times m}B_{n \times m}^T$

as we know, $B_{n \times m}B_{n \times m}^T$ is covarian matrix of matrix $B_{n \times m}$ and is symmetric.

Answer 1

Of course the sum of positive semidefinite matrices is positive semidefinite, $A_{n \times n}$ is positive semidefinite, and $B_{n \times m}B^\text{T}_{n \times m}$ is too since for all $x$ we have $$ x^\text{T} B_{n \times m} B^\text{T}_{n \times m} x = (B^\text{T}_{n \times m} x)^\text{T} B^\text{T}_{n \times m} x = \lVert B^\text{T}_{n \times m} x \rVert^2 \geq 0. $$ So, the matrix $C_{n\times n}$ is always positive semidefinite, too.