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Linear Systems and Least Squares and the LDLT decomposition A = LDLT , 8 D = 0 0 0 0 4 0 . 25 The diagonal elements in D are positive, and therefore we can put √ d1 √ d2 D1/2 = , .. . √ dn and then we get A = LDLT = (LD1/2 )(D1/2 LT ) = U T U, where U is an upper triangular matrix. This variant of the LDLT decomposition is called the Cholesky decomposition. Since A is symmetric, it is only necessary to store the main diagonal and the elements above it, n(n + 1)/2 matrix elements in all.

When the LDLT decomposition is computed, it is not necessary to first compute the LU decomposition, but the elements in L and D can be computed directly. 3 Perturbation Theory and Condition Number The condition number of a nonsingular matrix A is defined as κ(A) = A A−1 , where · denotes any operator norm. g. the 2-norm, then we write κ2 (A) = A 2 A−1 2 . 3) The condition number is used to quantify how much the solution of a linear system Ax = b is changed, when the matrix and the right hand side are perturbed by a small amount.

7) for the Frobenius norm and the identity tr(BC) = tr(CB). 2 Elementary Orthogonal Matrices We will use elementary orthogonal matrices to reduce matrices to compact form. For instance, we will transform a matrix A ∈ Rm×n , m ≥ n, to triangular form. 1 Plane rotations A 2 × 2 plane rotation matrix5 G= c −s s , c c2 + s2 = 1. ). Multiplication of a vector x by G rotates the vector in a clock-wise direction by an angle θ, where c = cos θ. A plane rotation can be used to zero the second element of a vector x by choosing c = x1 / x21 + x22 and s = x2 / x21 + x22 : 1 x21 + x22 x1 −x2 x2 x1 x1 x2 = x21 + x22 .