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Lecture Notes 21

# Lecture Notes 21 - Lecture21 Lecture 21 Towards the...

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Lecture21 1 Outline: 1) Properties of Symmetric Positive Definite Matrices (all eigenvalues >0) A) Tests B) Important PD matrices A T A and AA T 2) Overview of Eigenvalue factorizations AS=S Λ (generally) A=S Λ S -1 (diagonalizable) A=MJM -1 (non-diagonalizable) A=Q Λ Q T (symmetric) 3) The final Factorization, the incredible SVD Definition: A=U V T (or AV=U ) Mechanics: The Big Picture: The Applications: Total Least squares, image compression, EOF analysis Lecture 21: Towards the SVD (singular Value Decomposition) If A is real, square and symmetric: All Eigenvalues of A are _______________ All Eigenvectors of A can be chosen _________________ All symmetric matrices can be diagonalized Factorization: for A T =A, A=____________ Quick Review: Eigenvalues and Eigenvectors of Real Symmetric matrices Definitions: A is a positive definite (PD) matrix if A T =A and all its eigenvalues are > 0. If λ >= 0, A is said to be semi-positive definite . Examples: A=[ 1 2 ; 2 1 ], [ 1 2 ; 2 4], [ 1 2 ; 2 5] Positive Definite Matrices Quick Tests for PD: 1) All the pivots are positive A = [ 1 2 ; 2 d ] (Cholesky Factorization...) Positive Definite Matrices

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