diagonalizable

简明释义

英[/ˌdaɪəˈɡɒnəlaɪzəbl/]美[/ˌdaɪəˈɡɒnəlaɪzəbl/]

adj. 对角化的

英英释义

单词用法

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反义词

例句

1.Finally, a sufficient and necessary condition of the diagonalizable block compound matrices is proved.

最后让明了块复合矩阵可对角化的一个充要条件。

2.One kind of inverse eigenvalue problems, whose solutions are required to be normal or diagonalizable matrices, is investigated in quaternionic quantum mechanics.

本文研究了四元数量子力学中一类要求其解是正规或可对角化四元数矩阵的特征值反问题。

3.One kind of inverse eigenvalue problems, whose solutions are required to be normal or diagonalizable matrices, is investigated in quaternionic quantum mechanics.

本文研究了四元数量子力学中一类要求其解是正规或可对角化四元数矩阵的特征值反问题。

4.By decomposing a matrix into one diagonalizable matrix and two orthogonal matrixes, singular value decomposition has very good properties.

奇异值分解是将一矩阵分解为一个对角矩阵和两个正交矩阵，奇异值分解有着非常好的性质。

5.If a matrix has distinct eigenvalues, it is guaranteed to be diagonalizable.

如果一个矩阵具有不同的特征值，它一定是可对角化的。

6.A matrix is considered diagonalizable if it can be expressed in the form of a diagonal matrix, which makes calculations easier.

如果一个矩阵可以表示为对角矩阵的形式，那么它被认为是可对角化的，这使得计算更简单。

7.In linear algebra, not all matrices are diagonalizable, especially those that do not have enough eigenvectors.

在线性代数中，并非所有矩阵都是可对角化的，尤其是那些没有足够特征向量的矩阵。

8.The process of finding a diagonalizable matrix often involves calculating its characteristic polynomial.

找到一个可对角化的矩阵的过程通常涉及计算其特征多项式。

9.To determine if a matrix is diagonalizable, you can check its eigenvalues and their corresponding multiplicities.

要确定一个矩阵是否可对角化的，你可以检查它的特征值及其对应的重数。

作文

In the realm of linear algebra, one of the critical concepts that students encounter is the idea of a matrix being diagonalizable. A matrix is said to be diagonalizable if it can be expressed in the form of a diagonal matrix through a similarity transformation. This means that there exists an invertible matrix P and a diagonal matrix D such that A = PDP^(-1), where A is the original matrix. Understanding this concept is essential for solving various problems in mathematics and engineering, as it simplifies many calculations and provides deeper insights into the properties of linear transformations.To grasp the concept of diagonalizable matrices, one must first understand eigenvalues and eigenvectors. Eigenvalues are scalars that indicate how much a corresponding eigenvector is stretched or compressed during a linear transformation represented by the matrix. If a matrix has n linearly independent eigenvectors, it is diagonalizable. This property is significant because it allows us to transform complex linear operations into simpler ones, making computations more manageable.For instance, consider a 2x2 matrix A. If we can find two distinct eigenvalues, λ1 and λ2, along with their corresponding eigenvectors, v1 and v2, we can construct the matrix P using these eigenvectors as columns. The diagonal matrix D will then have the eigenvalues on its diagonal: D = diag(λ1, λ2). This transformation not only provides a clearer understanding of the matrix's behavior but also facilitates the computation of powers of the matrix, as D^n can be easily calculated by raising each diagonal entry to the power n.However, not all matrices are diagonalizable. A common example is the Jordan block, which arises when a matrix has repeated eigenvalues but lacks enough linearly independent eigenvectors. In such cases, while the matrix may still be brought to a form that reveals some structure, it cannot be simplified into a purely diagonal form. This limitation highlights the importance of studying the conditions under which a matrix is diagonalizable.The implications of diagonalizable matrices extend beyond theoretical mathematics; they are applicable in various fields, including physics, computer science, and statistics. For example, in quantum mechanics, the observables are represented by operators that can often be diagonalizable, allowing for easier interpretation of measurement outcomes. Similarly, in computer graphics, transformations such as rotations and scaling can be efficiently computed using diagonalizable matrices, enhancing performance in rendering scenes.In conclusion, the concept of a matrix being diagonalizable is foundational in linear algebra, providing a powerful tool for simplifying and understanding linear transformations. By recognizing the significance of eigenvalues and eigenvectors, one can determine whether a matrix is diagonalizable and leverage this property to solve complex problems across various domains. As students delve deeper into the world of linear algebra, mastering the concept of diagonalizable matrices will undoubtedly enhance their mathematical toolkit, enabling them to tackle real-world challenges with greater ease and efficiency.

在线性代数领域，学生们遇到的一个关键概念是矩阵的对角化。如果一个矩阵可以通过相似变换表示为对角矩阵，那么这个矩阵就被称为对角化。这意味着存在一个可逆矩阵P和一个对角矩阵D，使得A = PDP^(-1)，其中A是原始矩阵。理解这一概念对于解决数学和工程中的各种问题至关重要，因为它简化了许多计算，并提供了对线性变换性质的更深入的见解。要理解对角化矩阵的概念，首先必须了解特征值和特征向量。特征值是标量，指示在由矩阵表示的线性变换中，相应的特征向量被拉伸或压缩的程度。如果一个矩阵有n个线性独立的特征向量，它就是对角化的。这个性质是重要的，因为它允许我们将复杂的线性运算转换为简单的运算，从而使计算更易于管理。例如，考虑一个2x2矩阵A。如果我们能找到两个不同的特征值λ1和λ2，以及它们对应的特征向量v1和v2，我们可以使用这些特征向量作为列构造矩阵P。对角矩阵D将在其对角线上具有特征值：D = diag(λ1, λ2)。这种变换不仅提供了对矩阵行为的更清晰理解，还方便了矩阵幂的计算，因为D^n可以通过将每个对角元素提升到n次方轻松计算。然而，并不是所有的矩阵都是对角化的。常见的例子是约旦块，当一个矩阵具有重复的特征值但缺乏足够的线性独立特征向量时，就会出现这种情况。在这种情况下，虽然矩阵仍然可以转化为揭示某种结构的形式，但它不能简化为纯对角形式。这一限制突显了研究矩阵在何种条件下是对角化的重要性。对角化矩阵的影响超越了理论数学；它们在物理、计算机科学和统计等各个领域都有应用。例如，在量子力学中，观测量由可以对角化的算符表示，从而便于解释测量结果。同样，在计算机图形学中，旋转和缩放等变换可以通过对角化矩阵高效计算，从而提高渲染场景的性能。总之，矩阵对角化的概念是线性代数的基础，为简化和理解线性变换提供了一种强大的工具。通过认识特征值和特征向量的重要性，人们可以确定一个矩阵是否是对角化的，并利用这一性质解决各个领域的复杂问题。随着学生深入研究线性代数，掌握对角化矩阵的概念无疑将增强他们的数学工具箱，使他们能够更轻松高效地应对现实世界的挑战。