eigenvectors

简明释义

英[ˈaɪɡənˌvɛktəz]美[ˈaɪɡənˌvɛktərz]

n. [数]特征向量；本征矢量（eigenvector 的复数形式）

英英释义

单词用法

同义词

反义词

例句

1.According to the corresponding eigenvectors, the 3 principal components were explained as the anthropogenic factor, the temperature factor, and the precipitation factor.

根据3个主成分各自对应的特征向量特点，分别解释为人为因子、温度因子和降水因子并进行回归分析。

2.Eigenvalues and Eigenvectors. DVR Method.

本征值和本征矢量。DVR方法。

3.Using the negative eigenvalue theory and the infinite order perturbation theory, we calculate numerically the eigenvalues and the eigenvectors of the electronic states.

利用负本征值理论及无限阶微扰理论，对系统电子的本征值和本征态进行了数值计算。

4.This paper provides an efficient algorithm for finding all the eigenvalues and partial eigenvectors of a real matrix.

本文提供求实矩阵全部特征值及部分特征矢量的一个富有成效的算法。

5.The signal eigenvectors of vibration signal for the training were extracted by wavelet packet by the, and then used for neural network training.

对于用于训练学习的振动信号用小波包变换的方法对信号进行特征值提取得到信号的特征向量，并对神经网络系统进行训练。

6.In Section 4, numerical examples and their results are detailed, with some attention given to the resulting eigenvectors.

第四节详述了数值顺序举例以及结果，特别描述了一些产生向量特征的情况。

7.The principal components are derived from the eigenvectors 特征向量 of the covariance matrix.

主成分是从协方差矩阵的eigenvectors 特征向量中导出的。

8.In machine learning, we often use eigenvectors 特征向量 to reduce the dimensionality of data.

在机器学习中，我们常常使用eigenvectors 特征向量来减少数据的维度。

9.To solve the system of differential equations, we need to find the eigenvectors 特征向量 of the matrix.

要解这个微分方程组，我们需要找到矩阵的eigenvectors 特征向量。

10.In quantum mechanics, the state of a quantum system is described by eigenvectors 特征向量 of an operator.

在量子力学中，量子系统的状态由算符的eigenvectors 特征向量描述。

11.The stability of a system can be analyzed using the eigenvectors 特征向量 of its Jacobian matrix.

系统的稳定性可以通过其雅可比矩阵的eigenvectors 特征向量进行分析。

作文

In the field of linear algebra, the term eigenvectors refers to a special set of vectors associated with a square matrix. An eigenvector is a non-zero vector that changes only in scale when a linear transformation is applied to it. This concept is fundamental in various applications, including physics, engineering, and data science. Understanding eigenvectors can provide deep insights into the properties of linear transformations and matrices.To grasp the significance of eigenvectors, let us consider a simple example. Imagine a two-dimensional space where we have a matrix representing a transformation. When this matrix is applied to certain vectors, those specific vectors will either stretch or compress but will not change their direction. These unique vectors are what we call eigenvectors. The corresponding scales by which they stretch or compress are known as eigenvalues.The mathematical definition of eigenvectors stems from the equation: Ax = λx, where A is the square matrix, x is the eigenvector, and λ (lambda) is the eigenvalue. This equation states that when the matrix A acts on the vector x, the result is simply the vector x scaled by the factor λ. This property makes eigenvectors particularly useful in simplifying complex problems.One of the most significant applications of eigenvectors is in the field of principal component analysis (PCA), which is widely used in data reduction and feature extraction. In PCA, eigenvectors of the covariance matrix represent the directions of maximum variance in the data. By projecting the original data onto these eigenvectors, we can reduce the dimensionality of the dataset while preserving as much information as possible. This technique is invaluable in machine learning, allowing for more efficient algorithms and better performance.In physics, eigenvectors play a crucial role in quantum mechanics. The state of a quantum system can be represented by eigenvectors of an operator, which corresponds to measurable quantities like position or momentum. The eigenvalues associated with these eigenvectors represent the possible outcomes of measurements. This connection between eigenvectors and physical phenomena illustrates the profound impact of linear algebra on our understanding of the universe.Moreover, eigenvectors are essential in stability analysis of systems. In control theory, the stability of a system can often be determined by analyzing the eigenvectors of its system matrix. If all eigenvalues have negative real parts, the system is stable; if any eigenvalue has a positive real part, the system is unstable. Thus, eigenvectors provide critical insights into the behavior of dynamic systems.In conclusion, eigenvectors are a cornerstone of linear algebra with wide-ranging applications across different fields. Their ability to maintain direction under linear transformations makes them invaluable for simplifying complex problems, whether in data science, physics, or engineering. By mastering the concept of eigenvectors, one can unlock new perspectives on various scientific and mathematical challenges. As we continue to explore the intricacies of mathematics, the importance of eigenvectors will undoubtedly remain a pivotal topic of study and application.

在线性代数领域，术语特征向量指的是与方阵相关的一组特殊向量。特征向量是一个非零向量，当对其应用线性变换时，仅在尺度上发生变化。这个概念在物理、工程和数据科学等各种应用中都是基础的。理解特征向量可以为线性变换和矩阵的性质提供深刻的见解。为了掌握特征向量的重要性，让我们考虑一个简单的例子。想象一个二维空间，我们有一个表示变换的矩阵。当这个矩阵应用于某些向量时，这些特定的向量将会拉伸或压缩，但不会改变其方向。这些独特的向量就是我们所称的特征向量。与之对应的拉伸或压缩的比例被称为特征值。特征向量的数学定义源于方程：Ax = λx，其中A是方阵，x是特征向量，λ（lambda）是特征值。这个方程表明，当矩阵A作用于向量x时，结果只是向量x按因子λ缩放。这一特性使得特征向量在简化复杂问题中尤为有用。特征向量最重要的应用之一是在主成分分析（PCA）领域，该领域广泛用于数据降维和特征提取。在PCA中，协方差矩阵的特征向量表示数据中最大方差的方向。通过将原始数据投影到这些特征向量上，我们可以在尽可能保留信息的情况下降低数据集的维度。这一技术在机器学习中是无价的，能够实现更高效的算法和更好的性能。在物理学中，特征向量在量子力学中发挥着关键作用。量子系统的状态可以通过操作符的特征向量来表示，这对应于可测量的量，如位置或动量。这些特征向量相关的特征值表示测量的可能结果。特征向量与物理现象之间的这种联系说明了线性代数对我们理解宇宙的深远影响。此外，特征向量在系统的稳定性分析中也是必不可少的。在控制理论中，系统的稳定性通常可以通过分析其系统矩阵的特征向量来确定。如果所有特征值的实部均为负，则系统是稳定的；如果任何特征值的实部为正，则系统是不稳定的。因此，特征向量为动态系统的行为提供了关键的见解。总之，特征向量是线性代数的基石，在不同领域具有广泛的应用。它们在保持方向不变的同时进行线性变换，使其在简化复杂问题中极为宝贵，无论是在数据科学、物理还是工程领域。通过掌握特征向量的概念，人们可以在各种科学和数学挑战中解锁新的视角。随着我们继续探索数学的复杂性，特征向量的重要性无疑将仍然是一个研究和应用的关键主题。