Find an eigenvector v1 corresponding to one of the

Question

1. The goal of this problem is to find the eigenvalues and eigenvectors for the following matrix:

(a) Find the eigenvalues of B.

(b) Find an eigenvector v₁ corresponding to one of the eigenvalues.

(c) Find an eigenvector v₃ corresponding to the remaining eigenvalue.

(d) Calculate v₁ · v₃. Are v₁ and v₂ linearly independent, orthogonal, both, or neither?

2. The goal of this problem is to find the eigenvalues and choose orthogonal eigenvectors for the following matrix:

(a) Find the eigenvalues of C. (There are two distinct eigenvalues; one negative and one positive. The positive e'genvaltie appears as a dou¬ble root of the characteristic equation.)

(b) Find an eigenvector v₁ corresponding to the negative eigenvalue.

(c) Find an eigenvector v₃ corresponding to the positive eigenvalue. It should be orthogonal to v₁. Why?

(d) Find another cigenvector v₃ corresponding to the positive eigenvalue. Choose it to be orthogonal to both v₁ and v₂.