If we have $S$ positive semi-definite matrices $A_1,\dots, A_S$ then what is the largest matrix positive semi definite matrix C such that $A_s -C$ is also psd for all $s=1,\dotsc,S$?
By largest I mean in terms of a suitable norm.
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2Imagine these were just numbers and that "positive semi-definite" meant "$\geq 0$"; what would the largest C be? – jbowman Nov 19 '20 at 22:49
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@jbowman Because numbers are very special--they are totally ordered--as a bit of counterpoint to that suggestion, let $S=2$ and suppose the matrices are $$\pmatrix{1&0\0&\epsilon}\quad\pmatrix{\epsilon&0\0&1}$$ for some $0\le\epsilon\lt 1.$ It helps illustrate the fact that psd does not induce a total order on the symmetric matrices. – whuber Nov 19 '20 at 23:08
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@jbowman Equivalence of psd matrices in terms of numbers would the non-negative part of the real line. The numbers are naturally ordered there, however, psd matrices are not. – noirritchandra Nov 20 '20 at 23:06