1. Let G have Laplacian matrix2L =-1-12.īoomoONTOTOO A. Use a matrix calculator to find the eigenvalues of L; there should besome pairs of them that have the same value. List them in orderIt's fine (suggested, actually) that you use decimal approximationsrather than exact values.

Answered: A. Use a matrix calculator to find the…

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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Question

1. Let G have Laplacian matrix
2
L =
-1
-1
2.
īoomo
ONTOTOO

A. Use a matrix calculator to find the eigenvalues of L; there should be
some pairs of them that have the same value. List them in order
It's fine (suggested, actually) that you use decimal approximations
rather than exact values.

Expert Solution

Step 1

We need to use a matrix calculator to find the eigen values of the matrix , and since it is a $7 \times 7$ order matrix therefore it has 7 eigen values .

$[\begin{matrix} 2 & 0 & 0 & - 1 & - 1 & 0 & 0 \\ 0 & 2 & - 1 & 0 & - 1 & 0 & 0 \\ 0 & - 1 & 3 & 0 & 0 & - 1 & - 1 \\ - 1 & 0 & 0 & 3 & 0 & - 1 & - 1 \\ - 1 & - 1 & 0 & 0 & 3 & 0 & - 1 \\ 0 & 0 & - 1 & - 1 & 0 & 2 & 0 \\ 0 & 0 & - 1 & - 1 & - 1 & 0 & 3 \end{matrix}]$

On using the matrix calculator we get the following eigen values ,

$λ_{1} = 0 λ_{2} = 1.382 λ_{3} = 1.382 λ_{4} = 3.618 λ_{5} = 3.618 λ_{6} = 2.5858 λ_{7} = 5.4142$

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