Rectangular matrix

Definition

A rectangular matrix is a matrix with m rows and n columns where m ≠ n. This term is used to differentiate such matrices from square matrices, where the number of rows and columns are equal.

Properties

Rectangular matrices have several unique properties that distinguish them from other types of matrices.

Rank

The rank of a rectangular matrix is the maximum number of linearly independent rows or columns. It is an important property that is used in various fields of mathematics, including linear algebra and matrix theory.

Nullity

The nullity of a rectangular matrix is the dimension of its null space, which is the set of all vectors that when multiplied by the matrix, result in the zero vector.

Determinant

Unlike square matrices, rectangular matrices do not have a determinant. This is because the determinant is defined as a function that takes a square matrix and returns a scalar value.

Eigenvalues and Eigenvectors

Rectangular matrices do not have eigenvalues and eigenvectors in the traditional sense. However, they can have singular values and singular vectors, which are generalizations of eigenvalues and eigenvectors for non-square matrices.

Types of Rectangular Matrices

There are several types of rectangular matrices that are commonly used in mathematics and its applications.

Row and Column Matrices

Row and column matrices are rectangular matrices with only one row or one column, respectively. They are also known as row vectors and column vectors.

Zero Matrix

A zero matrix is a rectangular matrix in which all the elements are zero. It is denoted by the symbol 0.

Identity Matrix

An identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. While not a rectangular matrix, it is often used in operations involving rectangular matrices.

Applications

Rectangular matrices are used in a wide range of applications, including linear algebra, statistics, computer graphics, and machine learning.

Linear Algebra

In linear algebra, rectangular matrices are used to represent linear transformations that do not preserve dimensionality. They are also used in the solution of systems of linear equations.

Statistics

In statistics, rectangular matrices are used to represent data sets where the number of variables does not equal the number of observations.

Computer Graphics

In computer graphics, rectangular matrices are used to perform transformations such as scaling, rotation, and translation on 3D models.

Machine Learning

In machine learning, rectangular matrices are used to represent the weights in artificial neural networks.