Math Problem Statement

Show that the entries of powers of the adjacency matrix A(G)^k are sums over paths of length k from vi to vj, where each path is given the weight product of wij. Compute the adjacency matrix for the directed graph with vertices v1, v2 with w11 = w12 = w21 = 1 and w22 = 0. Diagonalize it and conclude a closed form solution for the Fibonacci numbers.

Solution

Let's go through each part of this problem step-by-step.

Part 1: Powers of the Adjacency Matrix

  1. Statement: Show that the entries of powers of the adjacency matrix A(G)kA(G)^k are sums over paths of length kk from viv_i to vjv_j, where each path is given the weight (i,j)pathwij\prod_{(i,j) \in \text{path}} w_{ij}.

    Explanation: For an adjacency matrix A(G)A(G) of a graph GG with vertices v1,v2,,vnv_1, v_2, \dots, v_n, each entry AijA_{ij} represents the weight of the edge from vertex viv_i to vertex vjv_j. When we compute A(G)kA(G)^k, we are essentially calculating paths of length kk between each pair of vertices, where the weight of a path is the product of the weights along that path.

    Proof Outline:

    • The (i,j)(i, j)-th entry of A(G)kA(G)^k represents the total weight of all paths of length kk from viv_i to vjv_j.
    • Each such path contributes a weight equal to the product of weights along its edges, (i,j)pathwij\prod_{(i,j) \in \text{path}} w_{ij}.
    • Summing over all such paths, the (i,j)(i, j)-th entry of A(G)kA(G)^k is the sum of these products for all paths of length kk from viv_i to vjv_j.

Part 2: Compute the Adjacency Matrix and Diagonalize It

  1. Compute the Adjacency Matrix: For the directed graph with vertices v1v_1 and v2v_2 and weights w11=w12=w21=1w_{11} = w_{12} = w_{21} = 1 and w22=0w_{22} = 0, the adjacency matrix AA is: A=(1110)A = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}

  2. Diagonalize the Matrix:

    • To diagonalize AA, we find its eigenvalues and eigenvectors.
    • The characteristic polynomial of AA is: det(AλI)=1λ11λ=λ2λ1=0\det(A - \lambda I) = \begin{vmatrix} 1 - \lambda & 1 \\ 1 & -\lambda \end{vmatrix} = \lambda^2 - \lambda - 1 = 0
    • Solving λ2λ1=0\lambda^2 - \lambda - 1 = 0 yields the eigenvalues: λ1=1+52andλ2=152\lambda_1 = \frac{1 + \sqrt{5}}{2} \quad \text{and} \quad \lambda_2 = \frac{1 - \sqrt{5}}{2}
    • The eigenvectors corresponding to these eigenvalues can be calculated, allowing us to express AA in the form PDP1PDP^{-1}, where DD is a diagonal matrix with λ1\lambda_1 and λ2\lambda_2 on the diagonal, and PP is the matrix of eigenvectors.
  3. Closed Form for Fibonacci Numbers:

    • The matrix powers AkA^k relate to Fibonacci numbers because the entries of AkA^k correspond to Fibonacci numbers.
    • Using the diagonalization A=PDP1A = PDP^{-1}, we find that Ak=PDkP1A^k = PD^k P^{-1}, and DkD^k contains the powers λ1k\lambda_1^k and λ2k\lambda_2^k.
    • This results in a closed-form expression for Fibonacci numbers FkF_k, often represented as: Fk=λ1kλ2k5F_k = \frac{\lambda_1^k - \lambda_2^k}{\sqrt{5}} where λ1=1+52\lambda_1 = \frac{1 + \sqrt{5}}{2} and λ2=152\lambda_2 = \frac{1 - \sqrt{5}}{2} are the roots of the characteristic polynomial.

Would you like more details on any specific part or further clarification?


Here are five related questions for deeper exploration:

  1. How do the eigenvalues of an adjacency matrix affect the powers of the matrix?
  2. Can adjacency matrices for other types of graphs generate sequences like the Fibonacci sequence?
  3. How does the diagonalization of a matrix simplify the computation of its powers?
  4. What is the significance of the eigenvalues in relation to Fibonacci numbers?
  5. How would the adjacency matrix change if different weights were assigned to the edges?

Tip: When dealing with recurrence relations, expressing them in matrix form and diagonalizing can often reveal closed-form solutions.

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Math Problem Analysis

Mathematical Concepts

Graph Theory
Linear Algebra
Matrix Powers
Eigenvalues and Eigenvectors

Formulas

Matrix multiplication for finding powers of matrices
Characteristic polynomial for finding eigenvalues
Closed-form solution for Fibonacci numbers: F_k = (λ1^k - λ2^k) / sqrt(5)

Theorems

Matrix diagonalization theorem
Properties of adjacency matrices

Suitable Grade Level

Undergraduate Mathematics