Let A be a nonsingular n n matrix, and suppose that A = L1D1U1 = L2D2U2, where L1 and L2

For each of the following matrices, compute the determinants of all the leading principal submatrices and use them to determine whether the matrix is positive definite: (a) 2 1 1 2 (b) 3 4 4 2 (c) 6 4 2 4 5 3 2 3 6 (d) 4 2 1 2 3 2 1 2 5

Let A be a 33 symmetric positive definite matrix and suppose that det(A1) = 3, det(A2) = 6, and det(A3) = 8. What would the pivot elements be in the reduction of A to triangular form, assuming that only row operation III is used in the reduction process?

Let A = 2 1 0 0 1 2 1 0 0 1 2 1 0 0 1 2 (a) Compute the LU factorization of A. (b) Explain why A must be positive definite.

For each of the following, factor the given matrix into a product LDLT , where L is lower triangular with 1s on the diagonal and D is a diagonal matrix: (a) 4 2 2 10 (b) 9 3 3 2 (c) 16 8 4 8 6 0 4 0 7 (d) 9 3 6 3 4 1 6 1 9

Find the Cholesky decomposition LLT for each of the matrices in Exercise 4.

Let A be an n n symmetric positive definite matrix. For each x, y Rn, define _x, y_ = xTAy Show that _ , _ defines an inner product on Rn.

Let A be a nonsingular n n matrix, and suppose that A = L1D1U1 = L2D2U2, where L1 and L2 are lower triangular, D1 and D2 are diagonal, U1 and U2 are upper triangular, and L1, L2, U1, U2 all have 1s along the diagonal. Show that L1 = L2, D1 = D2, and U1 = U2. [Hint: L1 2 is lower triangular and U1 1 is upper triangular. Compare both sides of the equation D1 2 L1 2 L1D1 = U2U1 1 .]

Let A be a symmetric positive definite matrix and let Q be an orthogonal diagonalizing matrix. Use the factorization A = QDQT to find a nonsingular matrix B such that BTB = A.

Let B be an mn matrix of rank n. Show that BTB is positive definite. 1

Let A be a symmetric n n matrix. Show that eA is symmetric and positive definite. 1

Show that if B is a symmetric nonsingular matrix, then B2 is positive definite. 1

Let A = 1 1 2 1 2 1 and B = 1 1 0 1 (a) Show that A is positive definite and that xTAx = xTBx for all x R2. (b) Show that B is positive definite, but B2 is not positive definite. 1

Let A be an n n symmetric negative definite matrix. (a) What will the sign of det(A) be if n is even? If n is odd? (b) Show that the leading principal submatrices of A are negative definite. (c) Show that the determinants of the leading principal submatrices of A alternate in sign. 1

Let A be a symmetric positive definite nn matrix. (a) If k < n, then the leading principal submatrices Ak and Ak+1 are both positive definite and, consequently, have Cholesky factorizations Lk LT k and Lk+1LT k+1. If Ak+1 is expressed in the form Ak+1 = Ak yk yT k k where yk Rk and k is a scalar, show that Lk+1 is of the form Lk+1 = Lk 0 xT k k and determine xk and k in terms of Lk , yk , and k . (b) The leading principal submatrix A1 has Cholesky decomposition L1LT 1 , where L1 = ( a11 ). Explain how part (a) can be used to compute successively the Cholesky factorizations of A2, . . . , An. Devise an algorithm that computes L2, L3, . . . , Ln in a single loop. Since A = An, the Cholesky decomposition of A will be LnLT n . (This algorithm is efficient in that it uses approximately half the amount of arithmetic that would generally be necessary to compute an LU factorization.)