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a_(ij)*a_(ji)&le;0, eigenvalues of A=(a_(ij))

Source: SEEMOUS 2011 P2

June 18, 2021

matrixeigenvalueslinear algebra

Problem Statement

Let

A=(a_{ij})

be a real

n\times n

matrix such that

A^n\ne0

and

a_{ij}a_{ji}\le0

for all

i,j

. Prove that there exist two nonreal numbers among eigenvalues of

A