4.F.1 Suppose the eigenvalues of a 3 × 3 matrix A are 2, 3/5, and 2/5 with corresponding eigenvec- tors 1 2 -3 vi = v3 = 34 v2 -3 -3 For this problem you don't need to know A!! a) Describe A(v1 + 2v2 + 3v3) as a linear combination of {v1, v2, v3}. -2 b) Let ro = -5 |. Describe ro as a linear combination of {v1, v2, v3}. 3 c) Find the solution of the equation Ik+1 = Axk for the specified ro (e.g. find an explicit formula for æµ in terms of the eigenvectors v¡). d) As k → 00, describe the behavior of æk. (Answer both the question: does lim æ exist? What can you say about the line through æk, or the angle between x and any other vectors.) e) As k → -00, describe the behavior of æk. (Same parenthetical.)

If matrix A is square of size n x n with a rank equal to 2 and n > 2. The rank of a matrix is equal to the number of non-zero eigenvalues. Can you write A as the product of two matrices B E Rnxq and C E RPX" where we require that p < n and q < n? You may assume that A is diagonalizable. Show how to construct a matrix B and C such that A = BC.

The matrix 10 -24 -24 -10 has eigenvalues 26 and -26, and corresponding eigenvectors (3,-2) 1 and (2,3) respectively, both of magnitude 1. √13 (a) Write down an orthogonal matrix P and a diagonal matrix D such that PT AP = D. A = 20 1 /13 (b) Use your answer to part (a) to find the standard form of the equation of the non-degenerate conic with equation 10.x² - 48ry - 10y² + 3x - 2y = 0. (c) Hence state whether the conic is an ellipse, a parabola or a hyperbola.

One of the eigenvalues of the matrix 1 (²2 C a is 2(c+1), and a corresponding eigenvector is O-1-2c 02 (2c+1) (c + 1) 3 0-2 (2+ c) 0 (-²). What is the value of a?