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tridiag-short

# tridiag-short - A Tridiagonal Matrix We investigate the...

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Unformatted text preview: A Tridiagonal Matrix We investigate the simple n × n real tridiagonal matrix: M = α β ... β α β ... β α β ... . . . . . . . . . . . . . . . ... α β ... β α β ... β α = α I + β 1 ... 1 1 ... 1 1 ... . . . . . . . . . . . . . . . ... 1 ... 1 1 ... 1 = α I + β 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 μ = α + βλ . 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 λ 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 λ = 2 c . Then = ( T- λ 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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tridiag-short - A Tridiagonal Matrix We investigate the...

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