eigenvalues

简明释义

英[ˈaɪɡənˌvæljuːz]美['ɛgən,væljuz]

n. [数] 特征值；[数] 本征值；固有值

英英释义

单词用法

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反义词

例句

1.The coefficient matrices in the linear combination are numerical matrices which depend on the eigenvalues of the matrix $F$.

该通解是一组自由参向量的显式线性表示，其系数阵是依赖于矩阵F的特征值的数值矩阵。

2.We have obtained the coupled-mode equation by using the eigenvalues and eigenvectors, and given a theoretical analysis.

利用固有值和固有矢量的方法解出了耦合模方程，并对其进行理论性分析。

3.By solving these equations, statistical expressions of expectation and mean square covariance about eigenvalues will be gotten.

求解方程组可得到结构特征值的均值和均方差的统计特征的数值表达式。

4.Finally, based on these characteristics, the eigenvalues and eigenvalues polynomials and its diagonal matrix were given.

最后，在分块反对称反循环矩阵性质的基础上，给出了其特征值和特征多项式以及相似对角阵。

5.Such technique pursues, through the study of the eigenvalues, the reduction of the dimensionality in the representation space.

这种方法是通过对特征值的研究，追求表征空间的维数压缩。

6.The energy eigenvalues and the squeezed state solutions are obtained by making use of algebraic diagonalization.

利用代数对角化方法，可得到压缩态形式的能量本征态和相应的能量本征值。

7.Then the dynamic stability of the helical equilibrium depends on the eigenvalues in the time domain.

则螺旋线平衡的动态稳定性由时间域的本征值判断。

8.Eigenvalues and Eigenvectors. DVR Method.

本征值和本征矢量。DVR方法。

9.The responses of all elements of FAS system are determined by Runge-Kutta method and all eigenvalues are solved by the Q-R method.

采用龙格—库塔法确定系统的响应过程和采用Q-R法求解系统的全部特征值。

10.The principal component analysis relies heavily on the eigenvalues 特征值 to reduce dimensionality.

主成分分析在很大程度上依赖于eigenvalues 特征值 来降低维度。

11.The eigenvalues 特征值 of a transformation can indicate whether it is stable or unstable.

变换的eigenvalues 特征值 可以表明它是稳定还是不稳定。

12.To solve the differential equation, we need to find the eigenvalues 特征值 of the associated matrix.

要解决这个微分方程，我们需要找到相关矩阵的eigenvalues 特征值。

13.In quantum mechanics, the eigenvalues 特征值 represent measurable quantities like energy levels.

在量子力学中，eigenvalues 特征值 代表可测量的量，如能级。

14.In linear algebra, the eigenvalues 特征值 of a matrix are crucial for understanding its properties.

在线性代数中，矩阵的eigenvalues 特征值 对于理解其属性至关重要。

作文

In the field of mathematics and engineering, the concept of eigenvalues (特征值) plays a crucial role in understanding linear transformations and the behavior of various systems. An eigenvalue is a scalar that indicates how much a corresponding eigenvector is stretched or compressed during a linear transformation. This concept is not only fundamental in theoretical mathematics but also has practical applications in several scientific and engineering disciplines.To grasp the significance of eigenvalues, it is essential to first understand what an eigenvector is. An eigenvector is a non-zero vector that changes by only a scalar factor when a linear transformation is applied to it. The relationship between eigenvalues and eigenvectors can be expressed mathematically with the equation Ax = λx, where A is a linear transformation represented by a matrix, x is the eigenvector, and λ (lambda) is the eigenvalue.The study of eigenvalues is particularly important in various applications such as stability analysis, vibration analysis, and quantum mechanics. For instance, in structural engineering, understanding the eigenvalues of a system can help engineers determine the natural frequencies of structures, which is vital for ensuring stability and safety under dynamic loads, such as earthquakes or wind forces.Moreover, in the field of data science, eigenvalues are used in principal component analysis (PCA), a technique employed to reduce the dimensionality of large datasets while preserving as much variance as possible. In PCA, the eigenvalues represent the amount of variance captured by each principal component, allowing researchers to identify the most significant features of the data.In physics, particularly in quantum mechanics, the concept of eigenvalues is central to the study of quantum states. The eigenvalues of an operator correspond to the measurable quantities of a physical system, such as energy levels. Understanding these eigenvalues helps physicists predict the behavior of particles and their interactions.Despite its abstract nature, the concept of eigenvalues has profound implications in real-world scenarios. For example, in control theory, the eigenvalues of a system’s state matrix can determine the stability of the system. If all eigenvalues have negative real parts, the system is stable; if any eigenvalue has a positive real part, the system will exhibit unstable behavior.In conclusion, eigenvalues (特征值) are a fundamental concept in various fields of study, including mathematics, engineering, data science, and physics. Their ability to provide insights into the behavior and characteristics of linear transformations makes them invaluable for both theoretical exploration and practical application. As we continue to explore complex systems and data, the understanding of eigenvalues will remain a key component in advancing our knowledge and capabilities in numerous disciplines.

在数学和工程领域，eigenvalues（特征值）的概念在理解线性变换和各种系统的行为中发挥着至关重要的作用。eigenvalue是一个标量，指示在进行线性变换时相应的特征向量被拉伸或压缩的程度。这个概念不仅在理论数学中是基础性的，而且在多个科学和工程学科中具有实际应用。要理解eigenvalues的重要性，首先必须了解什么是特征向量。特征向量是一个非零向量，当对其应用线性变换时，仅以标量因子改变。eigenvalues与特征向量之间的关系可以用方程Ax = λx表示，其中A是由矩阵表示的线性变换，x是特征向量，而λ（lambda）是eigenvalue。eigenvalues的研究在稳定性分析、振动分析和量子力学等多个应用中尤为重要。例如，在结构工程中，了解系统的eigenvalues可以帮助工程师确定结构的固有频率，这对于确保在动态载荷下（如地震或风力）保持稳定性和安全性至关重要。此外，在数据科学领域，eigenvalues用于主成分分析（PCA），这是一种用于降低大型数据集维度的技术，同时尽可能保留方差。在PCA中，eigenvalues表示每个主成分所捕获的方差量，使研究人员能够识别数据中最重要的特征。在物理学中，尤其是在量子力学中，eigenvalues的概念是研究量子态的核心。算子的eigenvalues对应于物理系统的可测量量，如能级。理解这些eigenvalues有助于物理学家预测粒子的行为及其相互作用。尽管其抽象性，eigenvalues的概念在现实场景中具有深远的影响。例如，在控制理论中，系统状态矩阵的eigenvalues可以决定系统的稳定性。如果所有eigenvalues的实部都为负，则系统是稳定的；如果任何eigenvalue的实部为正，则系统将表现出不稳定行为。总之，eigenvalues（特征值）是多个研究领域中的基本概念，包括数学、工程、数据科学和物理学。它们提供了对线性变换的行为和特征的洞察，使其在理论探索和实际应用中都具有不可或缺的价值。随着我们继续探索复杂系统和数据，对eigenvalues的理解将始终是推动我们在多个学科知识和能力发展的关键组成部分。