Zero matrix
Definition
A Zero matrix or null matrix is a matrix in which all the elements are zero. It is denoted by the symbol 0 or 0m×n, where m and n are the dimensions of the matrix. It is a special case of a constant matrix. The zero matrix is the additive identity in the set of all m×n matrices.
Properties
The zero matrix has several important properties that make it a fundamental object in linear algebra. These properties are a direct result of the definition of matrix addition and scalar multiplication.
Additive Identity
The zero matrix serves as the additive identity in the set of all m×n matrices. This means that for any m×n matrix A, the sum of A and the zero matrix is A. In mathematical notation, this is expressed as A + 0 = A.
Additive Inverse
The zero matrix is the additive inverse of itself. That is, the sum of a zero matrix and itself is another zero matrix. This is expressed mathematically as 0 + 0 = 0.
Multiplicative Nullity
When the zero matrix is multiplied by any matrix, the result is another zero matrix. This is true regardless of the order of multiplication. In mathematical notation, this is expressed as A * 0 = 0 and 0 * A = 0 for any matrix A.
Determinant
The determinant of a square zero matrix is zero. This is because the determinant is calculated as the product of the elements on the main diagonal, and in a zero matrix, all these elements are zero.
Applications
The zero matrix is used in several areas of mathematics and its applications. Some of these include:
Linear Algebra
In linear algebra, the zero matrix is used in the definition of linear independence. A set of vectors is said to be linearly independent if the only linear combination that gives the zero vector is the trivial one where all coefficients are zero.
Differential Equations
In the study of differential equations, the zero matrix is used in the definition of a homogeneous system of linear differential equations. Such a system is said to be homogeneous if it can be written in matrix form as AY = 0, where A is a matrix of coefficients, Y is a vector of unknown functions, and 0 is the zero matrix.
Computer Science
In computer science, the zero matrix is used in the representation of graphs. In an adjacency matrix representation of a graph, the entry in the i-th row and j-th column is zero if there is no edge between the i-th and j-th vertices.