An Interior-Point Method for Approximate Positive Semidefinite Completions
Computational Optimization and Applications
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It is a straightforward matrix calculation that if $\lambda$ is an eigenvalue of $A,x$ an associated eigenvector and $\alpha$ the set of positions in which $x$ has nonzero entries, then $\lambda$ is also an eigenvalue of the submatrix of $A$ that lies in the rows and columns indexed by $\alpha$. A converse is presented that is the most general possible in terms of the data we use. Several corollaries are obtained by applying the main result to normal and Hermitian matrices. These corollaries lead to results concerning the case of equality in the interlacing inequalities for Hermitian matrices, and to the problem of the relationship among eigenvalue multiplicities in various principal submatrices.