Unformatted text preview: = b + 2 √ ac cos jπ n + 1 ,j = 1 , 2 ,...,n. (a) Verify this for n = 1 and then for n = 2 by direct calculation. (b) Verify this for n = 4 using MATLAB’s eig function, with a =-1, b = 2, and c =-1. (c) Verify this for general n but with c = a (symmetric case) by using the fact that the n × n matrix of eigenvectors, V , has elements v ij = sin ijπ n +1 . [Hint: write out a general interior row of A v j = λ j v j , which involves just three terms (since the matrix is tridiagonal) and use the trigonometric identity sin( α ± β ) = sin α cos β ± cos α sin β to replace the two oﬀ-diagonal terms (the ones involving a and c ) with multiples of the diagonal term.]...
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- Spring '08
- Matrices, trigonometric identity sin, Mathematics E2101y Introduction, Constant-coeﬃcient Linear ODEs, particular inhomogeneous solution