Lecture Notes 16

Lecture Notes 16 - Lecture16 October 29, 2007 Introduction...

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Lecture16 1 October 29, 2007 Outline: 1) Introduction: Eigenvalues, Eigenvectors: Ax = λ x 2) An example 3) Motivation: The applications Iterative maps and matrix powers Dynamical Systems du /dt=Au 4) An Algorithm for finding eigenvalues and eigenvectors 5) More Examples 6) simple checks Tr(A), |A| Lecture 16: Introduction to eigensystems The course so far: Part 1: Ax =b leads to PA=LU Part 2: A T Ax =A T b leads to A=QR Part 3: Fundamental Equation is Ax = λ x where A is square nxn λ is an Eigenvalue and x is an Eigenvector special directions such that Ax behaves like scalar multiplication (slightly misleading equation, we need to solve for both λ and x) (Factorization is A=S Λ S -1 or A=Q Λ Q T) Introduction to Eigen Problems An Example: A=[ 1 2; 2 1] Eigenvalues and Eigenvectors Foreshadowing: an Application -- iterative maps A large number of numerical methods can be written as an iterative method x k+1 =Ax k i.e.

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This note was uploaded on 06/02/2010 for the course APMA APMA E3101 taught by Professor Spiegelman during the Fall '07 term at Columbia.

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Lecture Notes 16 - Lecture16 October 29, 2007 Introduction...

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