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Suppose that $A \in {\cal S}_+^n$ is a symmetric positive semidefinite matrix. Let $B = {\rm sign}(A)$, where the sign is taken elementwise. Is the resulting matrix $B$ always positive semidefinite?

If not, under what conditions can we say that $B \in {\cal S}_+^n$ ?

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up vote 4 down vote accepted

A small symmetric perturbation of the identity matrix is positive definite, but the corresponding sign matrix need not be positive semidefinite. For example, let $A$ be the 3-by-3 matrix with 1 on the diagonal, and, say, -1/100 in the other 6 entries.

(I have nothing to say about your second question.)

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A trivial case when $B$ is semidef is say when the original matrix is elementwise non-negative (and you define sign(0) = 1). In that case, $B=ee^T$.

In general, it might be hard to give a non-trivial set of sufficient conditions on $A$ so that $B$ is semidefinite.

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