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Unformatted text preview: A Tridiagonal Matrix We investigate the simple n n real tridiagonal matrix: M = ... ... ... . . . . . . . . . . . . . . . ... ... ... = I + 1 ... 1 1 ... 1 1 ... . . . . . . . . . . . . . . . ... 1 ... 1 1 ... 1 = I + T , where T is defined by the preceding formula. This matrix arises in many applications, such as n coupled harmonic oscillators and solving the Laplace equation numerically. Clearly M and T have the same eigenvectors and their respective eigenvalues are related by = + . Thus, to understand M it is sufficient to work with the simpler matrix T . Eigenvalues and Eigenvectors of T Usually one first finds the eigenvalues and then the eigenvectors of a matrix. For T , it is a bit simpler first to find the eigenvectors. Let be an eigenvalue (necessarily real) and V = ( v 1 , v 2 ,..., v n ) be a corresponding eigenvector. With hindsight it will be convenient to write = 2 c . Then = ( T I ) V =  2 c 1 ... 1 2 c 1 ... 1 2 c 1 ... . . . . . . . . . . . . . . . ... 2 c 1 ... 1 2 c 1 ... 1 2 c v 1 v 2 v 3 . . . v n 2 v n 1 v n =  2 cv 1 + v 2 v 1 2 cv 2 + v 3 . . . v k 1 2 cv k + v k + 1 . . . v n 2 2 cv n 1 + v n v n 1 2 cv n (1) Except for the first and last equation, these have the form v k 1 2 cv k + v k + 1 = . (2) 1 We can also bring the first and last equations into this same form by introducing new arti ficial variables v and v n + 1 , setting their values as zero: v = 0, v n + 1 = 0....
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This note was uploaded on 03/06/2012 for the course MATH 260 taught by Professor Staff during the Spring '12 term at UPenn.
 Spring '12
 STAFF

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