Introducing the Dimension Keeping Semi-Tensor Product
A new matrix product that accommodates non-square matrices in mathematical operations.
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Mathematics often deals with various types of numbers and shapes, and in this world of numbers, Matrices play an essential role. Matrices are rectangular arrays of numbers that can represent data or help solve problems across many fields, such as engineering, physics, and computer science. This article will introduce a new type of matrix product called the dimension keeping semi-tensor product (DK-STP).
What is DK-STP?
The dimension keeping semi-tensor product allows matrices of different dimensions to multiply in a way that keeps the dimensions consistent. This means that if two matrices are multiplied, the result will still fit within the same dimensional space. This idea can adjust traditional matrix rules for Non-square Matrices, which are matrices that do not have an equal number of rows and columns.
Importance of Matrices in Various Fields
Matrices are not just abstract concepts; they have practical applications in multiple areas. For example, in computer graphics, matrices can rotate, scale, and translate images. In systems control, matrices help in analyzing the behavior of systems over time. In data science, matrices are essential tools for organizing and processing large datasets.
Basic Matrix Operations
Before diving deeper into the DK-STP, it's essential to understand some standard operations with matrices. The most common operations include addition, subtraction, and multiplication.
Addition and Subtraction
When adding or subtracting matrices, they must have the same dimensions. This means they must have the same number of rows and columns. If two matrices meet this requirement, you simply add or subtract their corresponding elements.
Multiplication
Matrix multiplication is a bit more complex. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. This operation combines rows and columns from the matrices to produce a new matrix.
Extending Operations to Non-Square Matrices
Traditionally, many operations are restricted to square matrices. Square matrices are matrices with equal rows and columns. However, many real-world problems involve non-square matrices. The DK-STP is designed to extend matrix operations, allowing them to apply to non-square matrices, thereby enabling the solution of a broader range of problems.
Applications of DK-STP
The introduction of DK-STP opens the door to various applications. Here are some key areas:
Engineering
In engineering, systems often require the analysis of processes over time. Using DK-STP can help model these systems effectively, as it allows for different dimensions through matrix representation.
Computation and Algorithms
In computer science, DK-STP can help improve algorithms involving matrices. This can enhance the efficiency of computations, especially in algorithms that require handling matrices of varying dimensions.
Dynamic Systems
Dynamic systems, which change over time, can be modeled using DK-STP. This provides a way to understand how systems evolve under different conditions.
Control Theory
Control theory deals with the behavior of dynamical systems. DK-STP can provide easier and more robust ways to analyze systems that need to maintain stability as conditions change.
Advantages of DK-STP
The following are some advantages of using DK-STP:
Flexibility
The primary benefit of DK-STP is its flexibility. It allows for the use of non-square matrices, which means that analysts can work with a broader range of data and systems.
Consistency in Dimensions
DK-STP ensures that operations retain consistent dimensions, allowing for the straightforward interpretation of results without the need for constant adjustments.
Simplified Calculations
By extending existing matrix operations to non-square matrices, DK-STP simplifies calculations that would otherwise be complex and cumbersome.
Key Concepts Related to DK-STP
To fully grasp DK-STP, one must understand several related concepts.
Eigenvalues And Eigenvectors
In linear algebra, eigenvalues and eigenvectors are crucial for understanding transformations. DK-STP allows for the extension of these concepts to non-square matrices, enabling the analysis of a more extensive variety of systems.
Determinants
The determinant of a matrix provides insight into its properties, such as Invertibility. DK-STP also allows the calculation of determinants for non-square matrices, which can be beneficial in various applications.
Invertibility
Invertibility refers to the ability to reverse a matrix operation. DK-STP enhances understanding of when non-square matrices can be inverted, which is vital for solving equations in many fields.
Additional Properties of DK-STP
Ring Structure
DK-STP can be organized into a ring structure, which is a mathematical framework that allows for more complex operations. This organization enables researchers to analyze relationships between different matrices within the context of DK-STP.
Lie Algebra and Lie Group
A Lie algebra is an algebraic structure that helps in understanding continuous symmetries. The DK-STP can create a Lie algebra and a corresponding Lie group. This development enriches the mathematical landscape surrounding matrix operations.
Group Actions
In mathematics, group actions describe symmetries of objects. DK-STP allows for such actions on dimension-free spaces, enhancing the understanding and application of matrices across various dimensions.
Future Directions
The work on DK-STP is only beginning. Several avenues remain open for future research and exploration.
Exploring Non-Square General Linear Groups
The study of non-square general linear groups, which arise from DK-STP, can yield valuable insights into matrix properties. A deeper understanding of these groups can help in various applications.
Control Systems with Dimension Perturbation
Investigating how dimension changes affect control systems is another promising direction. This can lead to improved models that account for real-world variances.
Analytical Functions
Defining analytical functions for non-square matrices can be another future focus. Such definitions will expand the mathematical toolbox and enhance applications.
Conclusion
The development of the dimension keeping semi-tensor product (DK-STP) marks an important advancement in the study of matrices. By allowing for more flexibility and applicability of non-square matrices, DK-STP opens new doors in engineering, computation, dynamic systems, and control theory. As researchers delve deeper into its properties and applications, the potential for breakthroughs in many fields becomes excitingly apparent. Exploring this new mathematical landscape will undoubtedly lead to valuable discoveries and innovations in various domains.
Title: From DK-STP to Non-square General Linear Algebras and General Linear Groups
Abstract: A new matrix product, called dimension-keeping semi-tensor product (DK-STP), is proposed. Under DK-STP, the set of $m\times n$ matrices becomes a semi-group $G({m\times n},\mathbb{F})$, and a ring, denoted by $R(m\times n,\mathbb{F})$. Moreover, the Lie bracket can also be defined, which turns the ring into a Lie algebra, called non-square (or STP) general linear algebra, denoted by $\mathrm{gl}(m\times n, \mathbb{F})$. Then the action of semi-group $G(m\times n,\mathbb{F})$ on dimension-free Euclidian space, denoted by $\mathbb{R}^{\infty}$, is discussed. This action leads to discrete-time and continuous time S-systems. Their trajectories are calculated, and their invariant subspaces are revealed. As byproduct of this study, some important concepts for square matrices, such as eigenvalue, eigenvector, determinant, invertibility, etc., have been extended to non-square matrices. Particularly, it is surprising that the famous Cayley-Hamilton theory can also been extended to non-square matrices. Finally, a Lie group, called the non-square (or STP) general Lie group and denoted by $\mathrm{GL}(m\times n,\mathbb{F})$, is constructed, which has $\mathrm{GL}(m\times n,\mathbb{F})$ as its Lie algebra. Their relations with classical Lie group $\mathrm{GL}(m,\mathbb{F})$ and Lie algebra $\mathrm{gl}(m,\mathbb{F})$ are revealed.
Authors: Daizhan Cheng
Last Update: 2023-07-02 00:00:00
Language: English
Source URL: https://arxiv.org/abs/2305.19794
Source PDF: https://arxiv.org/pdf/2305.19794
Licence: https://creativecommons.org/licenses/by/4.0/
Changes: This summary was created with assistance from AI and may have inaccuracies. For accurate information, please refer to the original source documents linked here.
Thank you to arxiv for use of its open access interoperability.