MATRIX ALGEBRA (CONTINUE) The generalized inverse of a matrix Definition 10 A generalized inverse of a matric a is ano ther matric A+ that sat is fies LAATA=A 4. AA is summe Quadratic Forms and Definite Matrices For the opt imization problem =∑∑ The quadratic form can be written as Ax For a given matrix A 1. If x'Ax>(<)0 for all nonzero x, then A is positive(negative) definite 2. If x'Ax 2(5)0 for all nonzero x, then A is nonnegative definite or positive semi-definite Some useful results · If A is nonnegative definite,then|A|≥0. If A is posit ive definite, so is A The identity matrix I is positive definite If A is n x K with full rank and n> K, then A'A is positive definite and Aa is nonnegative definite If A is posit ive definite and B is a nonsingular matrix, then B'AB is positive definite If A is symmetric and idempotent, n xn with rank J, then every quadratic form in A can be written x AxMATRIX ALGEBRA (CONTINUE) 6 The Generalized Inverse of a Matrix Definition 10 A generalized inverse of a matrix A is another matrix A+ that satisfies 1. AA+A = A 2. A+AA+ = A+ 3. A+A is symmetric 4. AA+ is symmetric Quadratic Forms and Definite Matrices For the optimization problem q = n i=1 n j=1 xixjaij The quadratic form can be written as q = x ′Ax For a given matrix A, 1. If x ′Ax > (<)0 for all nonzero x, then A is positive (negative) definite. 2. If x ′Ax ≥ (≤)0 for all nonzero x, then A is nonnegative definite or positive semi—definite. Some useful results: • If A is nonnegative definite, then |A| ≥ 0. • If A is positive definite, so is A−1 . • The identity matrix I is positive definite. • If A is n × K with full rank and n > K, then A′A is positive definite and AA′ is nonnegative definite. • If A is positive definite and B is a nonsingular matrix, then B′AB is positive definite. • If A is symmetric and idempotent, n × n with rank J, then every duadratic form in A can be written x ′Ax = J i=1 y 2 i