Math Problem Statement
Solution
Let's go through each part of this problem step-by-step.
Part 1: Powers of the Adjacency Matrix
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Statement: Show that the entries of powers of the adjacency matrix are sums over paths of length from to , where each path is given the weight .
Explanation: For an adjacency matrix of a graph with vertices , each entry represents the weight of the edge from vertex to vertex . When we compute , we are essentially calculating paths of length between each pair of vertices, where the weight of a path is the product of the weights along that path.
Proof Outline:
- The -th entry of represents the total weight of all paths of length from to .
- Each such path contributes a weight equal to the product of weights along its edges, .
- Summing over all such paths, the -th entry of is the sum of these products for all paths of length from to .
Part 2: Compute the Adjacency Matrix and Diagonalize It
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Compute the Adjacency Matrix: For the directed graph with vertices and and weights and , the adjacency matrix is:
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Diagonalize the Matrix:
- To diagonalize , we find its eigenvalues and eigenvectors.
- The characteristic polynomial of is:
- Solving yields the eigenvalues:
- The eigenvectors corresponding to these eigenvalues can be calculated, allowing us to express in the form , where is a diagonal matrix with and on the diagonal, and is the matrix of eigenvectors.
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Closed Form for Fibonacci Numbers:
- The matrix powers relate to Fibonacci numbers because the entries of correspond to Fibonacci numbers.
- Using the diagonalization , we find that , and contains the powers and .
- This results in a closed-form expression for Fibonacci numbers , often represented as: where and 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:
- How do the eigenvalues of an adjacency matrix affect the powers of the matrix?
- Can adjacency matrices for other types of graphs generate sequences like the Fibonacci sequence?
- How does the diagonalization of a matrix simplify the computation of its powers?
- What is the significance of the eigenvalues in relation to Fibonacci numbers?
- 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
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