线性代数与图论：矩阵方法在图论中的应用

需积分: 10 28 浏览量更新于2023-07-05 收藏 973KB PDF 举报

"Graphs and Matrices" 是一本专注于图论与矩阵理论相互结合的书籍，作者通过线性代数和矩阵理论的方法探讨了图论中的重要结果。这本书旨在展示线性代数在图论研究中的核心作用，尽管在传统上，图论学者可能对使用线性代数持保留态度。书中的内容通常被归类于代数图论领域，参考了Biggs和Godsil及Royle的经典著作，但本书更加侧重于矩阵技术的应用，因此可以称为“线性代数图论”。在图论中，图是由顶点和边构成的数学结构，用于表示各种关系或网络。矩阵则是一种二维数组，常用于表示图的结构和性质。例如，邻接矩阵可以用来表示图中两个顶点之间是否存在边，而度矩阵记录了每个顶点的邻接边的数量。这些矩阵是分析图的连通性、循环、哈密顿路径等问题的关键工具。线性代数在图论中的应用包括但不限于以下方面： 1. **谱图论**：谱理论研究图的特征值和特征向量，这与图的性质紧密相关。例如，图的拉普拉斯矩阵的特征值可以提供关于图的连通性、奇环数和最小生成树的信息。 2. **图的分解**：矩阵分解如奇异值分解(SVD)和特征分解可用于图的分解，揭示图的结构信息，如图的聚类和社区结构。 3. **图的相似性和距离**：通过比较两个图的矩阵表示，可以计算它们之间的相似度或距离，这在图形数据挖掘和网络分析中有重要应用。 4. **遍历矩阵**：遍历矩阵用于描述随机游走过程在图上的行为，有助于理解网络的可达性和扩散过程。 5. **最优化问题**：线性规划和矩阵优化方法可以解决图论中的问题，如最小费用流、最大流和最小割。 6. **图的同构**：矩阵的等价性可以用来判断两个图是否同构，即它们在顶点重新排列后是否相同。 7. **动力系统和稳定性**：在动态网络中，矩阵常被用来描述系统的演化，其特征值和特征向量影响着系统的稳定性和动力学行为。 8. **图的生成树**：矩阵方法也可以用来寻找图的生成树，比如使用Prim算法或Kruskal算法的矩阵版本。此外，书中可能还会涵盖更复杂的主题，如图的谱半径、Fiedler向量（用于社区检测）、图的奇偶分解以及与图相关的矩阵的对角化问题。通过对这些概念的深入理解，读者能够掌握利用线性代数工具解决复杂图论问题的能力。这本书适合对图论和线性代数有基础了解，并希望进一步探索两者交叉领域的读者。

展开

1.2 Eigenvalues of symmetric matrices 5

1.2 Eigenvalues of symmetric matrices

Characteristic polynomial

Let A be an n ×n matrix. The determinant det(A −λ I) is a polynomial in the (com-

plex) variable λ of degree n and is called the characteristic polynomial of A. The

equation

det(A −λ I) = 0

is called the characteristic equation of A. By the fundamental theorem of algebra

the equation has n complex roots and these roots are called the eigenvalues of A.

We remark that it is customary to deﬁne the characteristic polynomial of A as

det(λ I −A) as well. This does not affect the eigenvalues.

The eigenvalues might not all be distinct. The number of times an eigenvalue

occurs as a root of the characteristic equation is called the algebraic multiplicity of

the eigenvalue.

We may factor the characteristic polynomial as

det(A −λ I) = (λ

−λ)···(λ

−λ).

The geometric multiplicity of the eigenvalue λ of A is deﬁned to be the dimension

of the null space of A −λ I. The geometric multiplicity of an eigenvalue does not

exceed its algebraic multiplicity.

If A and B are matrices of order m×n and n ×m, respectively, where m ≥ n, then

the eigenvalues of AB are the same as the eigenvalues of BA, along with 0 with a

(possibly further) multiplicity of m −n.

If λ

,..., λ

are the eigenvalues of A, then det A = λ

···λ

, while trace A =

+ ···+ λ

A principal submatrix of a square matrix is a submatrix formed by a set of rows

and the corresponding set of columns. A principal minor of A is the determinant of

a principal submatrix. A leading principal minor is a principal minor involving rows

and columns 1,..., k for some k.

The sum of the products of the eigenvalues, of A, taken k at a time, equals the

sum of the k ×k principal minors of A. When k = 1 this reduces to the familiar fact

that the sum of the eigenvalues equals the trace.

If λ

,..., λ

are the eigenvalues of the n×n matrix A, and if q(A) is a polynomial

in A, then the eigenvalues of q(A) are q(λ

),..., q(λ

If A is an n ×n matrix with the characteristic polynomial p(A), then the Cayley–

Hamilton theorem asserts that p(A) = 0. The monic polynomial q(A) of minimum

degree that satisﬁes q(A) = 0 is called the minimal polynomial of A.

Spectral theorem

A square matrix A is called symmetric if A = A

. The eigenvalues of a symmetric

matrix are real. Furthermore, if A is a symmetric n ×n matrix, then according to the

剩余181页未读，继续阅读

身份认证购VIP最低享 7 折!

30元优惠券

PowerupKit

粉丝: 0

线性代数与图论：矩阵方法在图论中的应用

graphs and matrices

Graphs and matrices

Graphs and Matrices.pdf

Adjacency matrices are symmetric for both directed and undirected graphs.

how to write an essay reporting graphs

how to write an essay to report graphs

数据结构知识图谱构建与可视化参考文献

which library can facilitate the process of machine learning in python?

bayesian networks and decision graphs pdf

最新资源