What is a matrix in algebra?
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Matrices can be generalized in different ways. Abstract algebra uses matrices with entries in more general fields or even rings, while linear algebra codifies properties of matrices in the notion of linear maps. It is possible to consider matrices with infinitely many columns and rows. Another extension are tensors, which can be seen as higher-dimensional arrays of numbers, as opposed to vectors, which can often be realised as sequences of numbers, while matrices are rectangular or two-dimensional arrays of numbers. Matrices, subject to certain requirements tend to form groups known as matrix groups. Similarly under certain conditions matrices form rings known as matrix rings. Though the product of matrices is not in general commutative yet certain matrices form fields known as matrix fields.
Matrix (mathematics)
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In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties. Some particular topics out of many include; operations defined on matrices (such as matrix addition, matrix multiplication and operations derived from these), functions of matrices (such as matrix exponentiation and matrix logarithm, and even sines and cosines etc. of matrices), and the eigenvalues of matrices (eigendecomposition of a matrix, eigenvalue perturbation theory).
Matrix analysis
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In mathematics, a matrix (plural: matrices) is a rectangular "array" of numbers, symbols, or expressions, arranged in "rows" and "columns". For example, the dimensions of the matrix below are 2 × 3 (read "two by three"), because there are two rows and three columns:
Provided that they have the same size (each matrix has the same number of rows and the same number of columns as the other), two matrices can be added or subtracted element by element (see Conformable matrix). The rule for matrix multiplication, however, is that "two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second" (i.e., the inner dimensions are the same, n for an ("m"×n)-matrix times an (n×"p")-matrix, resulting in an ("m"×"p")-matrix. There is no product the other way round, a first hint that matrix multiplication is not commutative. Any matrix can be multiplied element-wise by a scalar from its associated field.
Matrix (mathematics)
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The size of a matrix is defined by the number of rows and columns that it contains. A matrix with "m" rows and "n" columns is called an "m" × "n" matrix or "m"-by-"n" matrix, while "m" and "n" are called its "dimensions". For example, the matrix A above is a 3 × 2 matrix.
Matrix (mathematics)
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Familiar properties of numbers extend to these operations of matrices: for example, addition is commutative, that is, the matrix sum does not depend on the order of the summands: A + B = B + A.
The transpose is compatible with addition and scalar multiplication, as expressed by ("c"A) = "c"(A) and (A + B) = A + B. Finally, (A) = A.
"Multiplication" of two matrices is defined if and only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an "m"-by-"n" matrix and B is an "n"-by-"p" matrix, then their "matrix product" AB is the "m"-by-"p" matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B:
Matrix (mathematics)
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A submatrix of a matrix is obtained by deleting any collection of rows and/or columns. For example, from the following 3-by-4 matrix, we can construct a 2-by-3 submatrix by removing row 3 and column 2:
Matrix (mathematics)
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The determinant of a product of square matrices equals the product of their determinants: 
Adding a multiple of any row to another row, or a multiple of any column to another column, does not change the determinant. Interchanging two rows or two columns affects the determinant by multiplying it by −1. Using these operations, any matrix can be transformed to a lower (or upper) triangular matrix, and for such matrices the determinant equals the product of the entries on the main diagonal; this provides a method to calculate the determinant of any matrix. Finally, the Laplace expansion expresses the determinant in terms of minors, that is, determinants of smaller matrices. This expansion can be used for a recursive definition of determinants (taking as starting case the determinant of a 1-by-1 matrix, which is its unique entry, or even the determinant of a 0-by-0 matrix, which is 1), that can be seen to be equivalent to the Leibniz formula. Determinants can be used to solve linear systems using Cramer's rule, where the division of the determinants of two related square matrices equates to the value of each of the system's variables.
Matrix (mathematics)
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The projection matrix has a number of useful algebraic properties. In the language of linear algebra, the projection matrix is the orthogonal projection onto the column space of the design matrix formula_26.(Note that formula_27 is the pseudoinverse of X.) Some facts of the projection matrix in this setting are summarized as follows:
The projection matrix corresponding to a linear model is symmetric and idempotent, that is, formula_33. However, this is not always the case; in locally weighted scatterplot smoothing (LOESS), for example, the hat matrix is in general neither symmetric nor idempotent.
Projection matrix
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In mathematics, a matrix unit is an idealisation of the concept of a matrix, with a focus on the algebraic properties of matrix multiplication. The topic is comparatively obscure within linear algebra, because it entirely ignores the numeric properties of matrices; it is mostly encountered in the context of abstract algebra, especially the theory of semigroups.
Matrix unit
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In linear algebra, a real symmetric matrix represents a self-adjoint operator over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refers to one which has real-valued entries. Symmetric matrices appear naturally in a variety of applications, and typical numerical linear algebra software makes special accommodations for them.
Symmetric matrix
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In linear algebra, a real symmetric matrix represents a self-adjoint operator over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refers to one which has real-valued entries. Symmetric matrices appear naturally in a variety of applications, and typical numerical linear algebra software makes special accommodations for them.
Symmetry in mathematics
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The all-ones matrix arises in the mathematical field of combinatorics, particularly involving the application of algebraic methods to graph theory. For example, if "A" is the adjacency matrix of a "n"-vertex undirected graph "G", and "J" is the all-ones matrix of the same dimension, then "G" is a regular graph if and only if "AJ" = "JA". As a second example, the matrix appears in some linear-algebraic proofs of Cayley's formula, which gives the number of spanning trees of a complete graph, using the matrix tree theorem.
Matrix of ones
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In linear algebra, a Moore matrix, introduced by , is a matrix defined over a finite field. When it is a square matrix its determinant is called a Moore determinant (this is unrelated to the Moore determinant of a quaternionic Hermitian matrix). The Moore matrix has successive powers of the Frobenius automorphism applied to the first column, so it is an "m" × "n" matrix
Moore matrix
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In mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring or even a semiring. The matrix product is designed for representing the composition of linear maps that are represented by matrices. Matrix multiplication is thus a basic tool of linear algebra, and as such has numerous applications in many areas of mathematics, as well as in applied mathematics, statistics, physics, economics, and engineering. In more detail, if is an matrix and is an matrix, their matrix product is an matrix, in which the entries across a row of are multiplied with the entries down a column of and summed to produce an entry of . When two linear maps are represented by matrices, then the matrix product represents the composition of the two maps.
Matrix multiplication
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In mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization of the scalar logarithm and in some sense an inverse function of the matrix exponential. Not all matrices have a logarithm and those matrices that do have a logarithm may have more than one logarithm. The study of logarithms of matrices leads to Lie theory since when a matrix has a logarithm then it is in a Lie group and the logarithm is the corresponding element of the vector space of the Lie algebra.
Logarithm of a matrix