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Given an $n × n$ real symmetric matrix, with atleast two diagonal entries that are of opposite sign, does this matrix have both a positive eigenvalue and a negative eigenvalue?

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Yes, because the corresponding quadratic form is indefinite; it takes both positive and negative values.

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  • $\begingroup$ Right, if we take $y = e_k$, the quadratic form would be $a_{kk}$, if the real symmetric matrix was $A = [a_{ij}]_{i,j=1}^n$. $\endgroup$
    – Anu
    Jan 28, 2018 at 12:50

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