Find the eigenvalues A1 < A2 < A3 and associated unit eigenvectors ū1, ú2, ūz of the symmetric matrix 4 -2 -27 A = -2 -2 4 -2 4 -2 The eigenvalue A1 = has associated unit eigenvector ủ1 The eigenvalue d2 = has associated unit eigenvector ü2 =

The matrix A is a real symmetric 5 x 5 matrix with eigenvalues 3, 7, 10, 15, and 21. The vector xị is an eigenvector of A with eigenvalue 3, the vector x2 is an eigenvector of eigenvalue 10, the vector x4 is an eigenvector of A with eigenvalue 15, and the vector x; is an eigenvector of A with eigenvalue 21. with eigenvalue 7, the vector x3 is an eigenvector of A with Compute the dot products of the eigenvectors a; with each other. 21. x2 = I1. X3 = X1. 24 = x2 · X3 = I2 · X4 = X2 · X5 = 13 · X4 = 13 · X5 = X4: x5 =

Let M be a 2 × 2 matrix with eigenvalues ₁ = -1.2, A2 = 1 with corresponding eigenvectors Consider the difference equation with initial condition Xo = 5 3 V₁ = V2 = 2 xk+1 = Mxk Write the initial condition as a linear combination of the eigenvectors of M. That is, write x0 = c1V1 + C₂ V₂ = In general, xk= V1+ V2 ) * vi+ k ) v2 Specifically, x4 = For large k, xk≈ k

The matrix 4 0 0 A 0 4-8 -4 4 -8] has eigenvalues -4, 0, and 4. Find its eigenvectors. The eigenvalue -4 has associated eigenvector The eigenvalue 0 has associated eigenvector The eigenvalue 4 has associated eigenvector 9.