Let A be a 3x3 symmetric matrix. Assume that A has three eigenvalues: A₁ = -1, A₂ = 2, and A3 = 5. The vectors V₁ and V₂ given below, are eigenvectors of A corresponding, respectively, to X₁ and X₂: V1 = Enter the vector V3 in the form [C₁, C2, C3]: 0 2 V2 = Find a non-zero vector V3 which is an eigenvector of A corresponding to X3. -1

Let x = Express as a linear combination of 71, 72, and 73, and find Ax. Ax = be eigenvectors of the matrix A which correspond to the eigenvalues X₁ -1 -1 2 V₁ 2 = ₁+ 2 0 √₂+2 , V₂ = [2] , V3 = -0 x = == -1, A₂ = 2, and X3 = 4, respectively, and let -3 V3.

Let A be a symmetric n x n matrix. Let A1 and A2 be two eigenvalues with eigenvectors vị and v2 for A, respectively. Show that vi and v2 are orthogonal.

Consider the following matrix: A The following vectors are linearly independent eigenvectors of A: V1 = 2 -2 H -4 Diagonalize the matrix A, i.e. find an invertible matrix P and a diagonal matrix D such that A = PDP-1. 2 -6-8 2 V2 2-2 -8