Let A be a 3 × 3 real symmetric matrix and have eigenvalues λ1 = 2, λ2 = λ3 = 1. The eigenvectors corresponding to λ2 = λ3 = 1 are respectively given by v2=(1,0,1), v3=(1,2.-1),Find the eigenvector of A corresponding to λ1 = 2 and the matrix A

To find the eigenvalues, computedet3 − λ 0 0−3 4 − λ 90 0 3 − λ = (3 − λ)(4 − λ)(3 − λ).So the eigenvalues are λ = 3 and λ = 4.We can find two linearly independent eigenvectors301 ,130 corresponding to the eigenvalue 3, and oneeigenvector010 with eigenvalue 4. The diagonalized form of the matrix is3 0 0−3 4 90 0 3 =3 1 00 3 11 0 03 0 00 3 00 0 40 0 11 0 −3−3 1 9 .Note that if you chose different eigenvectors, your matrices will be different. The middle matrix should haveentries 3, 3, 4 in some order, and you should multiply out the product to make sure you have the right answer.

For each of the following matrices, and eigenvalue λ has been given.[ 0 4−1 5], λ = 4(a)[2 22 −1], λ = −2(b)3 1 −10 1 24 2 0, λ = 3(c)Find one eigenvector corresponding to the given eigenvalue

why can use this step:We can find the eigenvector corresponding to λ1 = 2 by taking the cross product of v2 and v3, since the cross product of two vectors gives a vector that is orthogonal to both. v1 = v2 x v3 = (1,0,1) x (1,2,-1) = (-2, 0, -2

A=( 3 2 4 2 0 2 4 2 3) the eigenvectors of A when λ=-1,it is (-1/2,1,0) and (-1,0,1) re the eigenvectors of A corresponding to the eigenvalue λ = −1 orthogonal with respect to the usual dot product? If not, apply the Gram-Schmidt process to turn them into an orthonormal set of eigenvectors

Let A=【3 2 4 2 0 2 4 2 3】 You are given that the characteristic equation of A is (λ + 1)2(λ − 8) = 0, i.e. the eigenvalues of A are λ = −1, 8. Are the eigenvectors of A corresponding to the eigenvalue λ = −1 orthogonal with respect to the usual dot product? If not, apply the Gram-Schmidt process to turn them into an orthonormal set of eigenvectors