For each of the following, factor the matrix A into a product QDQT where is orthogonal and D is diagonal. 1 (a) A Q: = = 00 -1888) -(888) D = (b) A = = -1 -1] -3 5 5 -3 8 -2 2 -2 5 -5 2 -5 5 D = 0 00 00

Solve each matrix equation for X

The matrix A = (13¹) is closely related to the imaginary number i. (a) Calculate A², A³, A4, expressing each as a multiple of either A or the identity matrix I. Infer the general pattern for A4, A4n+1, A4n+2 A4n+3, where n is an integer, and notice how this compares to powers of i (b) The exponential of a matrix is defined in terms of the power series eX = 1 + X + ² X ²X² + ²/1 ² Calculate e, evaluating the series to obtain a simple 2 x 2 matrix. Explain the result geometrically, noting the close analogy with e ·X³+.