Use the definition of convexity to show that $A$ is a convex set.

Question

Let $A$ be the set of all positive semidefinite definite $n × n$ matrices in $\mathbb R_{ n×n}$ . Use the definition of convexity to show that $A$ is a convex set.

Answer 1

Let $X,Y\in A$. Then we want to show that for all $t\in [0,1]$, $(1-t)A+tB$ is positive semi-definite. Let's show this by direct calculation. For any vector $x$: $$x^T((1-t)A + tB)x = (1-t)(x^TAx) + t(x^TBx)$$ Now you just need to show that the above is non-negative, using the fact that $A$ and $B$ are positive semi-definite.