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Lecture Notes 20 - Lecture20 Lecture 20 Eigenvalues and...

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Lecture20 1 Outline: 1) Big Idea: If A T =A then A=Q Λ Q T with Λ real always A) 3x3 example B) general 2x2 real eigenvalues C) general proof for all symmetric matrices 2) Properties of Symmetric matrices A) Spectral theorem B) sign of Pivots = signs of eigenvalues 3) Positive Definite Matrices (Symmetric, all eigenvalues >0) A) Tests B) Important PD matrices A T A and AA T 4) Similar Matrices and Jordan Canonical form Lecture 20: Eigenvalues and Eigenvectors of Symmetric Matrices Critical Properties of Eigenvalues and Eigenvectors: If A T =A 1) All eigenvalues are real 2) All eigenvectors can be chosen orthonormal 3) All symmetric matrices can be diagonalized 4) A=S Λ S -1 becomes A=Q Λ Q T Real Symmetric Matrices Example #1: A=[ 1 0 1; 0 2 0 ; 1 0 1] Real Symmetric Matrices Example #2: A=[ a b ; b d] (General symmetric 2x2 matrix) Real Symmetric Matrices
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Lecture20 2 Preliminaries: A useful identity for proving general properties of Eigenvalues and eigenvectors of Real Symmetric Matrices given A in R nxn and u ,v in R n ...
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