## A Problem with Negative Squares (Open)

Summary: Let $A$ be a block form symmetric real matrix of order $km$ in which each $A_{ij}$ is symmetric and of order $m$. The following question arises: if $A$ has $\kappa$ negative eigenvalues counting multiplicity, does the trace matrix $A_{11} + A_{22} + \cdots + A_{kk}$ have at most $\kappa$ negative eigenvalues? Show that the answer is "yes" for $k=8$.

Classification: Primary, Algebra; Secondary, Matrices and Determinants

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**Torben Maack Bisgaard**

Nandrupsvej 7 st. th.

DK-2000 Frederiksberg C

Denmark

e-mail: torben.bisgaard@get2net.dk