Chen
Aug9-08, 09:47 AM
Hi,
I'm reading a paper where the determinant of the following matrix is solved for using some kind of recurisve method.
The matrix is given by M_{ij} = A \delta_{i,j} - B \delta_{i,j-1} - C \delta_{i,j+1}, with i,j = 1...N and are NOT cyclic.
The author sets D_N = \texttt{det}\[M_{(N)}\] and writes the equation
D_N = A D_{N-1} - B C D_{N-2},
assumes a solution of the form D_N = \lambda ^N and finds two solutions,
\lambda_{\pm} = (A \pm \sqrt{A^2 - 4 B C}) / 2.
He then notes the initial conditions of D_1 = A and D_2 = A^2 - B C and says that the answer is therefore
D_N = \frac{\lambda_{+}^{N+1} - \lambda_{-}^{N+1}}{\lambda_{+} - \lambda_{-}}.
It's the very last step I don't understand, how did he find D_N?
Thanks
I'm reading a paper where the determinant of the following matrix is solved for using some kind of recurisve method.
The matrix is given by M_{ij} = A \delta_{i,j} - B \delta_{i,j-1} - C \delta_{i,j+1}, with i,j = 1...N and are NOT cyclic.
The author sets D_N = \texttt{det}\[M_{(N)}\] and writes the equation
D_N = A D_{N-1} - B C D_{N-2},
assumes a solution of the form D_N = \lambda ^N and finds two solutions,
\lambda_{\pm} = (A \pm \sqrt{A^2 - 4 B C}) / 2.
He then notes the initial conditions of D_1 = A and D_2 = A^2 - B C and says that the answer is therefore
D_N = \frac{\lambda_{+}^{N+1} - \lambda_{-}^{N+1}}{\lambda_{+} - \lambda_{-}}.
It's the very last step I don't understand, how did he find D_N?
Thanks