- #1
ognik
- 643
- 2
Homework Statement
Given an NxN symetric tri-diagonal matrix, derive the recursion relation for the characteristic polynomial Pn(λ)
Homework Equations
Pn(λ) = |A -λI |
Pn(λ) = (An,n - λ)Pn-1(λ) - A2n,n-1Pn-2(λ)
The Attempt at a Solution
This was easy to do by induction, but I am always looking to strengthen my index notation, so I'd like to also do it using index notation. Please also point out anything wrong with my notation.
Starting with the generalised $$ |A| = \sum_{i=1}^{N} {(-1)}^{i+j}{a}_{ij}{M}_{ij} $$
When i = j, aij = Aii - λ (Given)
For j = i ± 1, aij = aji (symetric)
For i+1 < j < i-1, aij = aji = 0 (tri-diagonal)
Since then I've been going round in circles, only achieving:
P1(λ) = A11 - λ (the recursion needs a starting value)
Expanding by the nth row (i=n, j from n to n-1 only, since for j=n-2, aij = 0):
$$ {P}_{n}\left(\lambda\right)=\sum_{i=n}^{i=n-1} {(-1)}^{i+j}{a}_{ij}{M}_{ij}= {\left(-1\right)}^{n+n}{a}_{n,n}{M}_{n,n}+{\left(-1\right)}^{n+n-1}{a}_{n,n-1}{M}_{n,n-1} $$
$$ =\left({A}_{n,n}-\lambda\right){M}_{n,n} - {a}_{n,n-1}{M}_{n,n-1} $$
I can see that I am almost there, the 2nd term looks similar to Pn-1, but I'm just stuck for the next move?