Recall that for an n x n matrix A, its characteristic polynomial Pa(x) is defined by PA(x) := det(xIn – A), | where I, denotes the n x n identity matrix. Consider the sequence of simple graphs Tn = (Vn, En) defined as follows for n > 0: Vn := {vo, . .. , Vn} and E, := {{v-1,t;} | i = 1,...,n}, with the convention that Eo = 0. Page 2 of 3 (a) Find the adjacency matrix A, of the graph T, for each n > 0. (b) Set Po(x) := 1, and for n > 1 let Pn(x) := det(xI, – An-1) be the characteristic polynomial of An-1. Find P1(x) and P2(x). (c) Show that for n > 2 we have that Pn(x) = xPn-1(x) – Pn–2(x). (Hint for part (c): consider the Laplace expansion of the determinant with respect to the first column of xIn – A,-1.)
Recall that for an n x n matrix A, its characteristic polynomial Pa(x) is defined by PA(x) := det(xIn – A), | where I, denotes the n x n identity matrix. Consider the sequence of simple graphs Tn = (Vn, En) defined as follows for n > 0: Vn := {vo, . .. , Vn} and E, := {{v-1,t;} | i = 1,...,n}, with the convention that Eo = 0. Page 2 of 3 (a) Find the adjacency matrix A, of the graph T, for each n > 0. (b) Set Po(x) := 1, and for n > 1 let Pn(x) := det(xI, – An-1) be the characteristic polynomial of An-1. Find P1(x) and P2(x). (c) Show that for n > 2 we have that Pn(x) = xPn-1(x) – Pn–2(x). (Hint for part (c): consider the Laplace expansion of the determinant with respect to the first column of xIn – A,-1.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:5. Recall that for an n x n matrix A, its characteristic polynomial PA(x) is defined by
PA(x) := det(xIn – A),
where In denotes the n x n identity matrix. Consider the sequence of simple graphs
Tn = (Vn, En) defined as follows for n 2 0:
Vn := {vo, . .., Vn}
and
En := {{vi-1, vi} | i = 1, ...,n},
with the convention that E, = Ø.
Page 2 of 3
(a) Find the adjacency matrix An of the graph Tn for each n > 0.
(b) Set Po(x) := 1, and for n > 1 let Pn(x) := det(xIm – An-1) be the characteristic
polynomial of A,–1. Find P(x) and P2(x).
(c) Show that for n > 2 we have that
Pn(x) = xPn-1(x) – Pn-2(x).
(Hint for part (c): consider the Laplace expansion of the determinant with respect to
the first column of xI, – An-1-)
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