Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $B \in {\cal S}_+^n$ ?

share|improve this question

2 Answers 2

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.)

share|improve this answer

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.