Lec18 - Eigenvalue Problems For Matrices and Boundary Value...

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Eigenvalue Problems For Matrices and Boundary Value Problems
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2 Linear Eigenvalue Problems z Eigenvalue problems are homogeneous , linear problems of the form: Matrix Eigenvalue Problem: BV Eigenvalue Problem, e.g. L=d 2 /dx 2 : z In both types of problems, there is a trivial solution, {y},y=0, but there are usually also a number of nontrivial solutions, {y n }, y n for particular (not arbitrary) values of λ = λ n , n=1, 2, … z These are called eigenvectors/eigenfunctions and eigenvalues
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3 Where do Eigenvalue Problems Arise? z Time-dependent linear systems: Assume: General solution: [A]: N x N matrix c n fixed by ICs
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4 Determination of Eigenvalues/Eigenvectors for Matrix Problems z Consider the Matrix Eigenvalue Problem: z By Cramer’s Rule, either {y}=0, or z For each λ n , solve linear system: “Characteristic equation ” – Nth order polynomial equation for the N eigenvalues Determines the N eigenvectors {y n }
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5 2x2 Matrix Example z Consider the matrix: z Characteristic equation is: z Eigenvector, n=1: z Eigenvector, n=2: Arbitrary to a constant factor
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6 Where Else do Eigenvalue Problems Arise? z Boundary Value Problems: e.g., Euler’s problem of a beam subjected to an axial load P E: elastic modulus I: moment of inertia of beam cross section P y x P (0,0) (L,0)
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7 Analytical Solution of Euler’s Problem General solution: Trivial solution: Eigenvalues and Eigenfunctions:
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Euler Buckling Loads z The eigenvalues and eigenfunctions of the Euler beam problem have the following significance: The beam is unstable to buckling to a shape given by the nth eigenfunction when the compressive load P exceeds P
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This note was uploaded on 12/29/2011 for the course CHE 132b taught by Professor Ceweb during the Fall '09 term at UCSB.

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Lec18 - Eigenvalue Problems For Matrices and Boundary Value...

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