Eigenvalues - MATH2071 LAB#12 The Eigenvalue Problem TABLE OF CONTENTS Introduction The Rayleigh Quotient The Power Method The Inverse Power Method

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MATH2071: LAB #12: The Eigenvalue Problem TABLE OF CONTENTS Introduction The Rayleigh Quotient The Power Method The Inverse Power Method Using Shifts The QR Method ASSIGNMENT Introduction For any square matrix A , consider the equation det(A-lambda*I)=0 . This is a polynomial equation of degree N in the variable lambda . Hence there are exactly N complex roots that satisfy the equation. If the matrix A is real, then the complex roots occur in conjugate pairs. The roots are known as eigenvalues of A . Interesting questions include: how can we find one or more of these roots? when are the roots distinct? when are the roots real? In textbook examples, the determinant is computed explicitly, the (usually cubic) equation is factored exactly, and the roots "fall out". This is not how a real problem will be solved. If lambda is an eigenvalue of A, then A-lambda*I is a singular matrix, and therefore there is at least one nonzero vector x with the property that (A-lambda*I)*x=0. This equation is usually written A * x = lambda * x Such a vector is called an eigenvector for the given eigenvalue. There may be as many as N linearly independent eigenvectors. In some cases, the eigenvectors are, or can be made, into a set of pairwise orthogonal vectors. Interesting questions about eigenvectors include: how do we compute an eigenvector? when will there be a full set of N independent eigenvectors? when will the eigenvectors be orthogonal? In textbook examples, the singular system (A-lambda*I)*x=0 is examined, and by inspection, an eigenvector is determined. This is not how a real problem is solved. Some useful facts about the eigenvalues lambda of a matrix A: A -1 has the same eigenvectors as A, and eigenvalues 1/lambda;
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A 2 has the same eigenvectors as A, and eigenvalues lambda 2 ; A+mu*I has the same eigenvectors as A, and eigenvalues lambda+mu; if A is symmetric, all its eigenvalues are real; if B is invertible, then inv(B)*A*B has the same eigenvalues as A; The Rayleigh Quotient If a vector x is an exact eigenvector of a matrix A, then it is easy to determine the value of the associated eigenvalue: simply take the ratio of the I-th components of A*x and x, for any index I that you like. But suppose x is only an approximate eigenvector, or worse yet, just a wild guess. We could still compute the ratio of corresponding components for some index, but the value we get will vary from component to component. We might even try computing all the ratios, and averaging them, or taking the ratios of the norms. However, the preferred method for estimating the value of an eigenvalue uses the Rayleigh quotient. The
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This note was uploaded on 04/26/2010 for the course CEG 616 taught by Professor Taylor during the Winter '10 term at Wright State.

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Eigenvalues - MATH2071 LAB#12 The Eigenvalue Problem TABLE OF CONTENTS Introduction The Rayleigh Quotient The Power Method The Inverse Power Method

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