Matrix multiplication
Introduction
Matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. Numbers such as the real or complex numbers can be multiplied according to elementary arithmetic. On the other hand, matrices are arrays of numbers, so there is no unique way to define "multiplication" of matrices. As such, in general the term "matrix multiplication" refers to a number of different ways to multiply matrices.
Basics of Matrix Multiplication
The most common and often used method of matrix multiplication is the so-called "dot product" method, also known as the "row-by-column" method. In this method, the elements of the rows of the first matrix are multiplied with the elements of the columns of the second matrix, and the results are summed up to get the elements of the resulting matrix.
Matrix Multiplication Properties
Matrix multiplication has a number of properties which are often useful in mathematical proofs and computations. These include:
- Associativity: For any three matrices A, B, and C, the equation (AB)C = A(BC) always holds true.
- Distributivity: For any three matrices A, B, and C, the equation A(B + C) = AB + AC always holds true.
- Identity: For any matrix A, there exists an identity matrix I such that IA = AI = A.
- Non-commutativity: In general, for any two matrices A and B, AB ≠ BA.
Matrix Multiplication in Linear Algebra
In linear algebra, matrix multiplication is one of the key operations. It can be used to represent a linear transformation, which is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.
Matrix Multiplication in Computer Science
In computer science, matrix multiplication is a fundamental operation in many areas such as graphics, physics simulation, and machine learning. Efficient algorithms for matrix multiplication are a subject of ongoing research.
Matrix Multiplication in Other Fields
Matrix multiplication also finds application in other fields such as physics, economics, and engineering. For example, in physics, matrices are used to describe the rotation of objects in three-dimensional space. In economics, matrices are used in input-output analysis to study the flow of goods and services in an economy.