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Given a symmetric positive semi-definite matrix $A = \{a_{ij}\}$, with $0 < a_{ij} < 1$, is the matrix $B = \{\frac{1}{1 - a_{ij}}\}$ also a symmetric positive semi-definite matrix?

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Yes. Since $|a_{ij}|<1$ for each $a_{ij}$, we may write $B$ as a convergent infinite series $I+A+A\circ A+A\circ A\circ A+\cdots$, where the symbol "$\circ$" denotes Hadamard product (i.e. entrywise product). Now the Schur product theorem states that the Hadamard product of two positive semidefinite matrices is also positive semidefinite. Hence each summand in the above infinite series is positive semidefinite and the sum is positive semidefinite too.

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The matrix

$$\begin{pmatrix}\frac1{10}&\frac1{51}\\ \frac12&\frac1{10}\end{pmatrix}\;\;\text{is positive semidefinite, but the matrix}\;\;\begin{pmatrix}\frac{10}9&\frac{51}{50}\\2&\frac{10}9\end{pmatrix}\;\;\text{is not...}$$

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