lecture18 - Differential Equations & Linear...

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Unformatted text preview: Differential Equations & Linear Algebra Lecture 18 Jianli XIE Email: xjl@sjtu.edu.cn Department of Mathematics, SJTU Fall 2010 Contents Eigenvalues and eigenvectors Eigenvalues and eigenvectors The equation Ax = y , can be viewed as a linear transformation that maps a given vector x into a new vector y . Vectors that are transformed into multiples of themselves are important in many applications. To find such vectors, we set y = λ x , where λ is a scalar proportionality factor, and seek solutions of the equation Ax = λ x , (1) or ( A- λ I ) x = . (2) The latter equation has nonzero solutions if and only if λ is chosen so that Eigenvalues and eigenvectors det( A- λ I ) = 0 . (3) Values of λ that satisfy equation (3) are called eigenvalues of the matrix A , and the nonzero solutions of (1) or (2) that are obtained by using such a value of λ are called the eigenvectors corresponding to that eigenvalue. To find all eigenvalues of a matrix, we can solve the equation (3) about λ , which is an n th order algebraic equation. Once an eigenvalue λ is found, we then solve system (2) for the corresponding eigenvectors. Remarks Since det( A- λ I ) is a polynomial of order n of the variable λ , the eigenvalues of a real matrix can be real or complex numbers. If x is an eigenvector corresponding to the eigenvalue λ , then any nonzero multiple of x is also an eigenvector, by equation (2). It is obvious that the eigenvalues of a diagonal matrix are its diagonal elements: if A = a 11 a 12 ··· a 1 n a 22 ··· a 2 n ....
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This note was uploaded on 05/17/2011 for the course ME 255 taught by Professor Jianlixie during the Fall '10 term at Shanghai Jiao Tong University.

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lecture18 - Differential Equations & Linear...

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