Review of matrix theory George H. Born Statistical Orbit Determination The University of Colorado, Boulder Fall, 2006 Lecture 5 Copyright 2006 Review of Matrix Concepts Copyright 2006 2 1 Review of Matrix Concepts Matrix Multiplication Copyright 2006 3 Review of Matrix Concepts Copyright 2006 4 2 Review of Matrix Concepts Copyright 2006 5 Review of Matrix Concepts Rank of a Matrix Copyright 2006 6 3 Review of Matrix Concepts Quadratic Forms Copyright 2006 7 Review of Matrix Concepts Quadratic Forms Copyright 2006 8 4 Review of Matrix Concepts A triangular matrix has non -zero elements on the diagonal and above (uppe triangular) or below (lower triangular). r Example of a u p p e r t r i a n g u l a r m a t r i x 3 !3 A unitary triangular matrix has diagonal elements equal to 1. Copyright 2006 &quot; a1 1 $ A=$ 0 $ 0 &amp; a1 2 a2 2 0 a1 3 # % a2 3 % a3 3 % ' 9 Review of Matrix Concepts The square root of an square root of P. m a t r i x , P, is defined as P=AA, where i s t h e A symmetric <a href="/keyword/positive-semidefinite/" >positive semidefinite</a> matrix has a unique symmetric <a href="/keyword/positive-semidefinite/" >positive semidefinite</a> square root. If P is symmet ric <a href="/keyword/positive-semidefinite/" >positive semidefinite</a> and then an orthogonal matrix Q exists so that n!n , w h e r e S is upper triangular, ST S = P AB ! Copyright 2006 A B 10 S = QA A= P 5 Review of Matrix Concepts Determinants Copyright 2006 11 Review of Matrix Concepts Determinants Copyright 2006 12 6 Review of Matrix Concepts Matrix Trace Copyright 2006 13 Review of Matrix Concepts Eigenvalues and Eigenvectors Copyright 2006 14 7 Review of Matrix Concepts Eigenvalues and Eigenvectors Copyright 2006 15 Review of Matrix Concepts Eigenvalues and Eigenvectors Example: Copyright 2006 16 8 Review of Matrix Concepts The Derivatives of Matrices and Vectors Copyright 2006 17 Review of Matrix Concepts The Derivatives of Matrices and Vectors Copyright 2006 18 9 Review of Matrix Concepts Maxima and Minima Copyright 2006 19 Review of Matrix Concepts Maxima and Minima Copyright 2006 20 10 Review of Matrix Concepts Maxima and Minima Copyright 2006 21 Review of Matrix Concepts Maxima and Minima Copyright 2006 22 11 Review of Matrix Concepts Maxima and Minima Copyright 2006 23 Review of Matrix Concepts Maxima and Minima Example: Copyright 2006 24 12 Review of Matrix Concepts Maxima and Minima Example: Copyright 2006 25 Review of Matrix Concepts Useful Matrix Inversion Theorems Copyright 2006 26 13 Review of Matrix Concepts Useful Matrix Inversion Theorems Copyright 2006 27 14