invertibility

简明释义

英[/ɪnˈvɜːrtəˌbɪlɪti/]美[/ɪnˈvɜːrtəˌbɪlɪti/]

n. [数] 可逆性

英英释义

单词用法

同义词

反义词

例句

1.This paper is a study on the invertibility of the commutator of scalar-idempotent elements and scalar-involutory elements of a unitary ring.

本文研究在一个有单位元的环中两个倍幂等元的换位子与两个倍对合元的换位子的可逆性问题。

2.This paper discusses the invertibility of single-input single output linear systems . A new method for construction the reduced inverse systems is derived.

本文对单输入单输出系统给出了可逆性的判断准则，建立了新的降阶逆系统的构造方法。

3.The structure of feedforward inverse finite automata is a fundamental problem in the invertibility theory of finite automata.

前馈逆有限自动机的结构是有限自动机可逆性理论中的基本问题。

4.In this paper, we discuss the stationarity and invertibility of a bilinear model.

本文主要讨论双线性时间序列模型的平稳性与可逆性。

5.This paper discusses the invertibility of single-input single output linear systems . A new method for construction the reduced inverse systems is derived.

本文对单输入单输出系统给出了可逆性的判断准则，建立了新的降阶逆系统的构造方法。

6.The invertibility of a function means that you can find its original input from its output.

函数的可逆性意味着你可以从输出中找到其原始输入。

7.In statistics, the invertibility of a covariance matrix is necessary for certain types of regression analysis.

在统计学中，协方差矩阵的可逆性对于某些类型的回归分析是必要的。

8.In control theory, the invertibility of a system's transfer function is important for designing feedback loops.

在控制理论中，系统传递函数的可逆性对于设计反馈回路非常重要。

9.The concept of invertibility is crucial in linear algebra, as it determines whether a matrix can be reversed.

在线性代数中，可逆性的概念至关重要，因为它决定了一个矩阵是否可以被反转。

10.The invertibility of a matrix is tested using the determinant; if it's zero, the matrix is not invertible.

矩阵的可逆性通过行列式来测试；如果为零，则该矩阵不可逆。

作文

In the realm of mathematics and linear algebra, the concept of invertibility plays a crucial role in understanding the behavior of linear transformations and matrices. To grasp the significance of invertibility, one must first understand what it means for a matrix to be invertible. A matrix is said to be invertible if there exists another matrix, known as its inverse, such that when the two matrices are multiplied together, they yield the identity matrix. This property is essential because it allows for the solution of linear equations and the manipulation of data in various applications.The importance of invertibility extends beyond pure mathematics; it finds relevance in fields such as computer science, engineering, and economics. For instance, in computer graphics, transformations such as rotation and scaling can be represented by matrices. If these transformation matrices are not invertible, it becomes impossible to reverse the transformation, which can lead to loss of information or distortion of the image.Moreover, in the context of systems of linear equations, the invertibility of a matrix determines whether a unique solution exists. When dealing with a system represented by the equation Ax = b, where A is a matrix, x is the vector of unknowns, and b is the output vector, the existence of a unique solution is guaranteed only if matrix A is invertible. If A is not invertible, the system may have no solutions or infinitely many solutions, complicating the analysis.From a practical standpoint, the concept of invertibility also applies to economic models. In economics, various models utilize matrices to represent relationships between different variables. The invertibility of these matrices is crucial for deriving meaningful conclusions from the models. For example, if an economist is trying to analyze the impact of changes in interest rates on inflation, the relationships can often be represented through matrix equations. If the corresponding matrix is not invertible, the analysis may yield ambiguous or misleading results.Furthermore, the concept of invertibility is not limited to square matrices. While only square matrices can possess inverses, rectangular matrices can still provide valuable insights through their rank and nullity. Understanding these properties can help in determining the conditions under which a system of equations can be solved, thus emphasizing the broader implications of invertibility in mathematical theory.In conclusion, invertibility is a fundamental concept in mathematics that has far-reaching implications across various disciplines. Its significance lies not only in theoretical applications but also in practical scenarios where decision-making relies on the ability to manipulate and interpret data effectively. By comprehending the principles of invertibility, one can better navigate the complexities of linear systems and their applications in real-world situations. As we continue to explore the interconnectedness of mathematics with other fields, the understanding of invertibility will remain a vital component of our analytical toolkit.

在数学和线性代数的领域中，invertibility（可逆性）这一概念在理解线性变换和矩阵的行为中起着至关重要的作用。要掌握invertibility的意义，首先必须理解一个矩阵可逆意味着什么。当存在另一个矩阵，即其逆矩阵，使得这两个矩阵相乘得到单位矩阵时，该矩阵就被称为可逆的。这个属性是至关重要的，因为它允许解决线性方程和在各种应用中操纵数据。invertibility的重要性超越了纯数学，它在计算机科学、工程和经济学等领域中也具有相关性。例如，在计算机图形学中，旋转和缩放等变换可以用矩阵表示。如果这些变换矩阵不是可逆的，那么就无法逆转变换，这可能导致信息丢失或图像失真。此外，在处理线性方程组时，矩阵的invertibility决定了是否存在唯一解。当处理由方程Ax = b表示的系统时，其中A是矩阵，x是未知向量，b是输出向量，只有当矩阵A是可逆的时，才保证存在唯一解。如果A不是可逆的，系统可能没有解或有无限多解，从而使分析变得复杂。从实际的角度来看，invertibility的概念也适用于经济模型。在经济学中，各种模型利用矩阵来表示不同变量之间的关系。这些矩阵的invertibility对于从模型中得出有意义的结论至关重要。例如，如果经济学家试图分析利率变化对通货膨胀的影响，这些关系通常可以通过矩阵方程来表示。如果相应的矩阵不可逆，则分析可能会产生模糊或误导性的结果。此外，invertibility的概念并不仅限于方阵。虽然只有方阵可以拥有逆，但矩形矩阵仍然可以通过其秩和零空间提供有价值的见解。理解这些属性可以帮助确定方程组可以解决的条件，从而强调invertibility在数学理论中的更广泛影响。总之，invertibility是数学中的一个基本概念，在各个学科中具有深远的影响。它的重要性不仅体现在理论应用中，也体现在依赖于有效操纵和解释数据的决策场景中。通过理解invertibility的原理，人们可以更好地应对线性系统及其在现实世界中的应用的复杂性。随着我们继续探索数学与其他领域的相互关联，invertibility的理解将始终是我们分析工具箱中的一个重要组成部分。