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.
Use the definition of convexity to show that $A$ is a convex set.
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convex-analysis
convex-optimization
nonlinear-optimization
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0Any nonnegative combination of "nonnegative things" (numbers, psd matrices, etc.) is again nonnegative... There is really nothing to show here. Just write it out. – 2017-02-13
1 Answers
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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.