List of named matrices

This page lists some important classes of matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called "entries". Matrices have a long history of both study and application, leading to diverse ways of classifying matrices. A first group is matrices satisfying concrete conditions of the entries, including constant matrices. Important examples include the identity matrix given by
and the zero matrix of dimension formula_2. For example:
Further ways of classifying matrices are according to their eigenvalues, or by imposing conditions on the product of the matrix with other matrices. Finally, many domains, both in mathematics and other sciences including physics and chemistry, have particular matrices that are applied chiefly in these areas.
Constant matrices.
The list below comprises matrices whose elements are constant for any given dimension (size) of matrix. The matrix entries will be denoted "aij". The table below uses the Kronecker delta δ"ij" for two integers "i" and "j" which is 1 if "i" = "j" and 0 else.
Specific patterns for entries.
The following lists matrices whose entries are subject to certain conditions. Many of them apply to "square matrices" only, that is matrices with the same number of columns and rows. The main diagonal of a square matrix is the diagonal joining the upper left corner and the lower right one or equivalently the entries "a""i","i". The other diagonal is called anti-diagonal (or counter-diagonal).
Matrices satisfying some equations.
A number of matrix-related notions is about properties of products or inverses of the given matrix. The matrix product of a "m"-by-"n" matrix "A" and a "n"-by-"k" matrix "B" is the "m"-by-"k" matrix "C" given by
This matrix product is denoted "AB". Unlike the product of numbers, matrix products are not commutative, that is to say "AB" need not be equal to "BA". A number of notions are concerned with the failure of this commutativity. An inverse of square matrix "A" is a matrix "B" (necessarily of the same dimension as "A") such that "AB" = "I". Equivalently, "BA" = "I". An inverse need not exist. If it exists, "B" is uniquely determined, and is also called "the" inverse of "A", denoted "A"−1.
Matrices used in statistics.
The following matrices find their main application in statistics and probability theory.
Matrices used in graph theory.
The following matrices find their main application in graph and network theory.