(a) For a symmetric matrix A consider the modified matrix B given by Al B = A- "volv(1).T This is referred to as Hotelling deflation. Suppose A has eigenvalues X1, A2, ..., An and eigenvectors v(1), .... v(#). i. Show that B has the same eigenvectors v(1), ... . v(). ii. Show that B has the eigenvalues A2, ..., An and 0. (b) For a matrix A consider the modified matrix B given by B = A-Xiv(1)x7, where x is chosen so that x v() =1. Consider the choice with 1 X = (1) (ail, . .., Win) ; for any index : with v," * 0. The aj = [A] are entries of the matrix A. This is referred to as Wielandt deflation. i. Show the ith row of B is zero, bix = 0 for k = 1, . . .n. ii. Show the it^ component of any eigenvector w of B is zero, w; = 0. iii. Show that B has the eigenvalues A2, ..., An and 0