# 2013-04 Largest eigenvalue of a symmetric matrix

Let $$H$$ be an $$N \times N$$ real symmetric matrix. Suppose that $$|H_{kk}| < 1$$ for $$1 \leq k \leq N$$. Prove that, if $$|H_{ij}| > 4$$ for some $$i, j$$, then the largest eigenvalue of $$H$$ is larger than $$3$$.

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