Matrix multiplication, or matrix product, is the act of multiplying two matrices together. It is an important operation used in mathematics, applied sciences, and computer programming. This article explores the properties of matrix multiplication and helps you understand Matrix operations. It also introduces complexities and performance considerations that should be taken into account when working with matrices and matrix multiplication. Finally, this article provides some examples in which matrix multiplication has been and can be applied.
What is Matrix Multiplication?
Matrix multiplication is the operation of multiplying two matrices together. This operation involves a set of rules which dictate when and how to calculate values of the resulting matrix. The rules of matrix multiplication are used to determine whether or not two matrices can be multiplied together and how the resulting matrix should be related to the two source matrices.
Matrix multiplication is an important operation in linear algebra and is used in many applications, such as solving systems of linear equations, calculating the inverse of a matrix, and finding the determinant of a matrix. It is also used in many areas of mathematics, such as calculus, statistics, and graph theory.
The Basics of Matrix Multiplication
Matrix multiplication involves both linear algebra and a series of axioms or properties. As a result, understanding matrix multiplication requires familiarity with basic linear algebra concepts such as vectors and matrices. To multiply two matrices together, the number of columns in the first matrix should match the number of rows in the second matrix. In traditional matrix notation, the two source matrices are written side-by-side. Each element in the resulting matrix is a linear combination of elements in its respective row in the first source matrix and corresponding column in the second source matrix.
The order of the matrices is important when multiplying them together. If the order is reversed, the result will be different. Additionally, matrix multiplication is not commutative, meaning that the order of the matrices matters. It is also important to note that matrix multiplication is not associative, meaning that the order of the matrices matters even when multiplying more than two matrices together.
Understanding the Properties of Matrix Multiplication
To understand matrix multiplication it is important to think not only about the numbers being operated on but also about the axioms governing their operations. There are several axioms associated with matrix multiplication that must be taken into account when working with matrices. These include the commutative, associative, distributive, identity, inverse, and trace properties.
Commutative Property of Matrix Multiplication
The commutative property states that reversing the order of factors does not affect the result of the multiplication. This property holds for matrix multiplication, meaning that A*B=B*A for two matrices A and B, as long as their dimensions are compatible.
Associative Property of Matrix Multiplication
The associative property states that the order in which three or more factors are multiplied does not influence the result. This property holds for matrix multiplication as well, meaning that (A*B)*C = A*(B*C) for three compatible matrices A, B, and C.
Distributive Property of Matrix Multiplication
The distributive property states that multiplying a sum of two numbers is equal to adding the products of multiplying each number individually. Again, this property holds for matrix multiplication, meaning that (A+B)*C = A*C+B*C for three compatible matrices A, B, and C.
Identity Property of Matrix Multiplication
The identity property states that multiplying a number by 1 gives back that number. This holds for matrix multiplication as well, where an identity matrix I, whose dimensions match those of A, when multiplied by A gives back A. This can be written as I*A=A.
Inverse Property of Matrix Multiplication
The inverse property states that multiplying a number by its inverse (1/n) gives back 1. Again, this holds for matrix multiplication as well. If A is an invertible matrix (i.e., there exists some other matrix A-1 such that A * A-1 = I), then A-1*A=I. An example of an invertible matrix is a diagonal matrix D whose elements are non-zeroes (D11, D22…Dnn).
Trace Property of Matrix Multiplication
The trace property states that when multiplying matrices of the same dimension a single value is produced by summing up certain elements of the source matrices. This is called a trace. Traces are commonly used to detect faulty programming when working with matrices. The trace of two compatible matrices A and B is defined as tr(A*B) = tr(B*A) = Σ(Aij*Bji).
Complexity and Performance Considerations for Matrix Multiplication
Matrix multiplication is a computationally expensive process due to its large data sets and diverse operations. Different algorithms and techniques can be used to reduce the time required to operate on big matrices. Understanding complexities and having a thorough understanding of performance tradeoffs are essential to successful matrix operations.
Applications and Examples of Matrix Multiplication
Matrix multiplication can be used for various applications, including graphics processing, image manipulation and analysis, computer-aided design (CAD), and machine learning algorithms (such as Artificial Neural Networks). An example of how matrix multiplicaiton can be used in graphic processing can be seen in 3D graphics modeling packages such as Blender. In this application, translations between coordinate systems are implemented using matrix multiplications.
