Question

Diagonalize the following matrix. -15 8 123565 204812 The result of multiplying your previous re sult is 70665 110492

Answer 1

A = [-3 -15 2 8] The Eigen values of A are the roots of the roots of the characteristics det(A - lambda I) = 0 So, det([-3 -15 2 8] - lambda [1 0 0 1]) = 0 So, det([-3 - lambda -15 2 8 - lambda]) = 0 So, (-3 - lambda) (8 - lambda) + 30 = 0 So, lambda^2 - 5 lambda + 6 = 0 So, (lambda - 3) (lambda - 2) = 0: lambda = 3, 2 The diagonal matrix D is composed of the Eigen values: D = [3 0 0 2] Find Eigen vectors: Solve the system (A - lambda I)x^vector = o^vector for lambda = 3: (A - 3I) x^vector = o^vector So, ([-3 -15 2 8] - 3[1 0 0 1]) [x y] = [0 0] So, [-6 -15 2 5] [x y] = [0 0] Reduce the matrix: apply R_2 leftarrow R_2 - 5/2 R_1 and R_1 leftarrow -1/6 R_1 So, [1 0 -1/3 0] [x y] = [0 0] So, x - 1/3y