# 2018-12 Property of Eigenvectors

Let $$A$$ be a $$2\times 2$$ matrix. Prove that if $$Av_1=\lambda_1v_1$$ and $$Av_2=\lambda_2v_2$$ for distinct reals $$\lambda_1$$ and $$\lambda_2$$ and nonzero vectors $$v_1$$ and $$v_2$$, then both columns of $$A-\lambda_1 I$$ is a multiple of $$v_2$$.

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# 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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# 2009-21 Rank and Eigenvalues

Let A=(aij) be an n×n matrix such that aij=cos(i-j)θ and θ=2π/n. Determine the rank and eigenvalues of A.

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