eigenvalue

简明释义

英[ˈaɪɡənˌvæljuː]美[ˈaɪɡənˌvæljuː]

n. [数] 特征值

英英释义

单词用法

同义词

反义词

例句

1.Traditional eigenvalue analysis method can not reveal the nonlinear interaction of power system.

传统的特征根分析法不能揭示电力系统的非线性相关作用。

2.Eigenvalue problem of a class of elliptic operators is studied.

研究了一类椭圆型算子的特征值问题。

3.The exact solution of derivatives of eigenvalue and eigenvector is presented by utilizing the mathematical theorem and new definitions of sensitivities.

本文利用数学定理和新的灵敏度定义给出特征值和特征向量导数的精确解。

4.In some cases, some local eigenvalue of the standard data could confirm the position.

有时由局部的标准特征也可以确定被测件的位置。

5.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。

6.In this paper, two kinds of structure inverse eigenvalue problems are discussed.

本文主要讨论了两类结构矩阵的逆特征值问题。

7.Last, I present some numerical results for one dimensal and two dimensal eigenvalue problems.

最后分别对一维二维问题给出了数值算例，其结果与理论相符。

8.In linear algebra, the eigenvalue 特征值 of a matrix represents a scalar that indicates how much a corresponding eigenvector is stretched or squished.

在线性代数中，矩阵的eigenvalue 特征值表示一个标量，指示相应的特征向量被拉伸或压缩的程度。

9.In quantum mechanics, the eigenvalue 特征值 corresponds to measurable quantities like energy levels.

在量子力学中，eigenvalue 特征值对应于可测量的量，如能量水平。

10.In data analysis, principal component analysis relies on the eigenvalue 特征值 to reduce dimensionality.

在数据分析中，主成分分析依赖于eigenvalue 特征值来降低维度。

11.Finding the eigenvalue 特征值 of a transformation can reveal important information about its stability.

寻找变换的eigenvalue 特征值可以揭示有关其稳定性的重要信息。

12.The eigenvalue 特征值 decomposition of a matrix can help us understand its properties more easily.

矩阵的eigenvalue 特征值分解可以帮助我们更容易地理解其性质。

作文

In the realm of mathematics and physics, the concept of eigenvalue plays a crucial role in understanding linear transformations and systems of equations. An eigenvalue, which can be translated into Chinese as '特征值', is a scalar that indicates how much a corresponding eigenvector is stretched or compressed during a linear transformation. This relationship is fundamental in various applications, including stability analysis, quantum mechanics, and even in data science through techniques such as Principal Component Analysis (PCA).To grasp the significance of eigenvalues, let us consider a simple example involving a square matrix. Suppose we have a matrix A that represents a linear transformation in a two-dimensional space. When we apply this transformation to a vector, it can change the vector's direction and magnitude. However, certain special vectors, known as eigenvectors, remain in the same direction after the transformation, albeit scaled by the eigenvalue.Mathematically, this relationship can be expressed with the equation A*v = λ*v, where A is the matrix, v is the eigenvector, and λ (lambda) is the eigenvalue. This equation tells us that when the matrix A acts on the eigenvector v, the result is simply the eigenvector scaled by the eigenvalue λ. This property is not only fascinating but also immensely useful in simplifying complex problems in linear algebra.In practical scenarios, eigenvalues help in determining the stability of systems. For instance, in control theory, the stability of a system can often be assessed by examining the eigenvalues of its system matrix. If all the eigenvalues have negative real parts, the system is stable; if any eigenvalue has a positive real part, the system is unstable. This principle is vital for engineers and scientists who design systems that need to operate reliably under various conditions.Moreover, in quantum mechanics, eigenvalues are associated with observable quantities. When a quantum system is measured, the possible outcomes correspond to the eigenvalues of the operator representing the observable. The eigenvalue gives us the value that can be observed, while the associated eigenvector provides the state of the system before measurement. This connection between eigenvalues and physical observables highlights their importance in theoretical physics.In the field of data science, eigenvalues are used in dimensionality reduction techniques. PCA, for instance, transforms high-dimensional data into lower dimensions while preserving as much variance as possible. The eigenvalues obtained from the covariance matrix of the data indicate the amount of variance captured by each principal component. By selecting the components with the largest eigenvalues, analysts can reduce the complexity of the data while retaining essential information.In conclusion, the concept of eigenvalue is integral to various fields of study, from mathematics to engineering and physics. Understanding eigenvalues allows us to analyze systems more effectively, predict behaviors, and simplify complex problems. As we continue to explore the depths of mathematics and its applications, the significance of eigenvalues will undoubtedly remain a cornerstone of our understanding.

在数学和物理的领域中，eigenvalue（特征值）这一概念在理解线性变换和方程组方面起着至关重要的作用。eigenvalue是一个标量，它表示在进行线性变换时，相关的特征向量被拉伸或压缩的程度。这种关系在各种应用中都是基础，包括稳定性分析、量子力学，甚至在数据科学中通过主成分分析（PCA）等技术。为了理解eigenvalues的重要性，我们可以考虑一个涉及方阵的简单例子。假设我们有一个矩阵A，它表示二维空间中的线性变换。当我们对一个向量应用这个变换时，它可能会改变向量的方向和大小。然而，某些特殊的向量，即特征向量，在变换后仍保持相同的方向，只是被eigenvalue拉伸或缩放。从数学上讲，这种关系可以用方程A*v = λ*v来表示，其中A是矩阵，v是特征向量，λ（lambda）是eigenvalue。这个方程告诉我们，当矩阵A作用于特征向量v时，结果仅仅是特征向量被eigenvalue λ缩放。这一特性不仅迷人，而且在简化复杂问题的线性代数中极为有用。在实际场景中，eigenvalues有助于确定系统的稳定性。例如，在控制理论中，系统的稳定性通常可以通过检查其系统矩阵的eigenvalues来评估。如果所有的eigenvalues的实部为负，则系统是稳定的；如果任何一个eigenvalue的实部为正，则系统是不稳定的。这一原则对于设计需要在各种条件下可靠运行的系统的工程师和科学家至关重要。此外，在量子力学中，eigenvalues与可观察量相关。当测量量子系统时，可能的结果对应于表示可观察量的算子的eigenvalues。eigenvalue给我们提供了可以观察到的值，而相关的特征向量则提供了测量前系统的状态。这种eigenvalues与物理可观察量之间的联系突显了它们在理论物理中的重要性。在数据科学领域，eigenvalues用于降维技术。例如，PCA将高维数据转换为低维数据，同时尽可能保留方差。从数据的协方差矩阵获得的eigenvalues指示每个主成分所捕获的方差量。通过选择具有最大eigenvalues的成分，分析人员可以减少数据的复杂性，同时保留重要信息。总之，eigenvalue的概念在多个研究领域中是不可或缺的，从数学到工程和物理。理解eigenvalues使我们能够更有效地分析系统，预测行为，并简化复杂问题。随着我们继续探索数学及其应用的深度，eigenvalues的重要性无疑将继续成为我们理解的基石。