eigenvector

简明释义

英[ˈaɪɡənvektə(r)]美[ˈaɪdʒənˌvektər]

n. [数] 特征向量；本征矢量

英英释义

单词用法

同义词

反义词

例句

1.Google's PageRank is a variant of the Eigenvector centrality measure.

谷歌的网页级别是特征向量中心性度量的一个变体。

2.Linear transformations and linear operators, eigenvector expansion, matrices as representations of linear operators.

线性转换及线性运算子，特征值扩展，以矩阵表示线性运算子。

3.In the appendix, the general method of finding eigenvalue and eigenvector is stated for reference.

在附录中还列出特征值与特征矢量的一般求法，以资与正文所述相对照。

4.Matrix iteration method can be employed to figure out the first eigenvalue and eigenvector of a matrix.

矩阵迭代法是求矩阵的第一阶特征值与特征向量的一种数值方法。

5.In this method, the image is divided into several modules firstly, then these sub-images' eigenvector is extracted by using PCA.

在对图像进行特征提取前，首先对图像进行划分，然后再对子图像使用PC A方法提取特征向量。

6.By using the orthogonality condition of augmented eigenvector of multibody system, the algebraic equation group with unknowns of structure parameter is formed.

利用多体系统增广特征矢量的正交性条件，可建立以系统物理参数为未知数的代数方程组。

7.State matrix is diagonalized according to the orthonormality of left and right eigenvector series about the state matrix.

最后根据状态矩阵的左右特征向量系的正交性，对给出的状态矩阵进行对角化。

8.The use of elementary geometry method can give conformal demonstration of lineal function in infinite eigenvector.

用初等几何方法给出线性函数在无穷远点的保角性证明。

9.The eigenvector 特征向量 corresponding to the largest eigenvalue indicates the direction of maximum variance.

与最大特征值对应的eigenvector 特征向量 表示最大方差的方向。

10.In quantum mechanics, the state of a system can be represented by an eigenvector 特征向量 of an operator.

在量子力学中，系统的状态可以由一个算符的eigenvector 特征向量表示。

11.To solve the linear transformation, we need to find the eigenvector 特征向量 of the matrix.

要解决线性变换，我们需要找到矩阵的eigenvector 特征向量。

12.In principal component analysis, we often seek the eigenvector 特征向量 that captures the most variance in the dataset.

在主成分分析中，我们通常寻找能够捕捉数据集中最大方差的eigenvector 特征向量。

13.Each eigenvector 特征向量 in a system can represent a different mode of vibration.

系统中的每个eigenvector 特征向量可以代表不同的振动模式。

作文

In the realm of linear algebra, the concept of an eigenvector is fundamental to understanding the behavior of linear transformations. An eigenvector, by definition, is a non-zero vector that changes only in scale when a linear transformation is applied to it. This means that when we apply a matrix to an eigenvector, the output is simply a scalar multiple of the original vector. Mathematically, this can be expressed as A*v = λ*v, where A is a square matrix, v is the eigenvector, and λ (lambda) is a scalar known as the eigenvalue.To grasp the significance of eigenvectors, consider their application in various fields such as physics, computer science, and statistics. In physics, eigenvectors are used to describe the states of quantum systems. For example, in quantum mechanics, the state of a particle can be represented by an eigenvector of an observable operator, which helps in predicting the outcomes of measurements. This relationship between eigenvectors and physical states illustrates how they serve as a bridge between abstract mathematical concepts and real-world phenomena.In computer science, particularly in machine learning, eigenvectors play a crucial role in techniques like Principal Component Analysis (PCA). PCA is a method used for dimensionality reduction, which simplifies the complexity of data while retaining its essential features. By identifying the eigenvectors of the covariance matrix of a dataset, we can determine the directions in which the data varies the most. These eigenvectors correspond to the principal components, allowing us to project high-dimensional data into a lower-dimensional space without losing significant information.Moreover, eigenvectors are essential in the field of graph theory. They help in analyzing the properties of graphs through the study of their adjacency matrices. The eigenvectors of these matrices can reveal important information about the structure of the graph, such as connectivity and clustering tendencies. For instance, in social network analysis, eigenvectors can be used to identify influential nodes within a network, thus providing insights into the dynamics of social interactions.The importance of eigenvectors extends to engineering as well. In control theory, eigenvectors are utilized to analyze the stability of systems. By examining the eigenvectors of a system's state matrix, engineers can determine whether a system will converge to a stable state or diverge over time. This application is crucial in designing systems that require reliable performance, such as in aerospace and automotive engineering.In conclusion, the concept of an eigenvector is not just a theoretical construct; it has practical implications across various disciplines. Its ability to represent fundamental properties of linear transformations makes it a powerful tool in mathematics and science. Understanding eigenvectors allows us to unlock deeper insights into complex systems, whether they be physical, computational, or social in nature. As we continue to explore the vast applications of eigenvectors, it becomes evident that they are integral to the advancement of knowledge in numerous fields, bridging the gap between abstract theory and practical application.

在线性代数的领域中，特征向量的概念对于理解线性变换的行为至关重要。根据定义，特征向量是一个非零向量，当应用线性变换时，仅在规模上发生变化。这意味着当我们将一个矩阵应用于一个特征向量时，输出只是原始向量的标量倍数。从数学上讲，这可以表示为 A*v = λ*v，其中 A 是一个方阵，v 是 特征向量，λ（lambda）是一个称为特征值的标量。为了理解特征向量的重要性，可以考虑它们在物理、计算机科学和统计等各个领域的应用。在物理学中，特征向量用于描述量子系统的状态。例如，在量子力学中，粒子的状态可以由可观测算子的特征向量表示，这有助于预测测量的结果。这种特征向量与物理状态之间的关系说明了它们如何在抽象数学概念和现实现象之间架起桥梁。在计算机科学中，特别是在机器学习中，特征向量在主成分分析（PCA）等技术中发挥着关键作用。PCA是一种用于降维的方法，它简化了数据的复杂性，同时保留其本质特征。通过识别数据集协方差矩阵的特征向量，我们可以确定数据变化最大的方向。这些特征向量对应于主成分，使我们能够在不失去重要信息的情况下将高维数据投影到低维空间。此外，特征向量在图论领域也至关重要。它们通过研究图的邻接矩阵来帮助分析图的属性。这些矩阵的特征向量可以揭示有关图结构的重要信息，例如连通性和聚类倾向。例如，在社交网络分析中，特征向量可以用来识别网络中的影响节点，从而提供对社交互动动态的洞察。特征向量的重要性还扩展到工程领域。在控制理论中，特征向量用于分析系统的稳定性。通过检查系统状态矩阵的特征向量，工程师可以确定系统是会收敛到稳定状态还是随时间发散。这一应用在设计需要可靠性能的系统时至关重要，例如在航空航天和汽车工程中。总之，特征向量的概念不仅仅是一个理论构造；它在各个学科中具有实际意义。它代表线性变换的基本属性，使其成为数学和科学中的强大工具。理解特征向量使我们能够深入洞察复杂系统，无论这些系统是物理的、计算的还是社会的。随着我们继续探索特征向量的广泛应用，显而易见它们在推动多个领域知识进步中是不可或缺的，架起了抽象理论与实际应用之间的桥梁。