The openness of the set of positive definite square matrices

matrices functional-analysis normed-spaces

Solution 1:

Restricting to the unit ball is always illustrating. Let $A$ be a given positive definite matrix, then there is $\delta>0$ such that \begin{equation} <Ax,x>\ge\delta \end{equation} for all $\|x\|=1$.

We use the 2-norm, defined by \begin{equation} \|A\|=\operatorname{sup}_{\|x\|=1}\|Ax\|, \end{equation} which is equivalent to any other norms.

If $B$ is very close to $A$, say, $\|B-A\|<\epsilon$, then \begin{equation} |<Bx,x>-<Ax,x>|=|<(B-A)x,x>|<\epsilon\|x\|^2, \end{equation} so if you restrict to the unit ball again then you can bound $<Bx,x>$ from below using positive definiteness of $A$ and controlling $\epsilon$, and this will lead to the positive definiteness of $B$.

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