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 \(v_2\).
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 \( 3 \).
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.
Let A be a 0-1 square matrix. If all eigenvalues of A are real positive, then those eigenvalues are all equal to 1.