"In mathematics, a matrix is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object."
Basic matrix operations include addition, multiplication, and scalar multiplication. More advanced operations include inversion, determinant, and eigenvalues.
Matrices: A matrix is a rectangular array of numbers or elements, arranged in rows and columns, enclosed in square brackets.
Operations on Matrices: Matrix addition, subtraction, scalar multiplication are some of the types of operations performed on matrices.
Matrix Multiplication: Matrix multiplication is a binary operation that produces a matrix from two matrices.
Determinants: A determinant is a scalar value that can be calculated for square matrices. It is used to determine whether a matrix has an inverse or not.
Inverse Matrices: An inverse matrix is a matrix that when multiplied by the original matrix, gives the identity matrix.
Eigenvalues and Eigenvectors: Eigenvalues and Eigenvectors are used to describe the behavior of a matrix, and how it affects other matrices.
Transposes: The transpose of a matrix is formed by exchanging rows for columns.
Row Echelon Form: The row echelon form of a matrix is a special form that can be used to simplify solving systems of linear equations.
Reduced Row Echelon Form: Reduced row echelon form is a further reduction of the row echelon form which is useful for solving systems of linear equations.
Singular Value Decomposition: Singular value decomposition is a factorization of a matrix that is used in numerous fields like image processing, signal processing, and statistics.
LU Decomposition: LU Decomposition is a matrix factorization technique that is used for solving systems of linear equations.
QR Decomposition: QR Decomposition is a matrix factorization technique that is used to solve least-squares problems and systems of linear equations.
Rank of a Matrix: Rank is a measure of the dimension of the column space of a matrix. It is useful in solving systems of linear equations and for determining whether a matrix is invertible or not.
Null Space: Null Space is the set of all solutions to the homogeneous system of linear equations.
Orthogonal and Orthonormal Bases: Orthogonal and Orthonormal Bases are used in matrix factorization, system-solving, and computation of eigenvalues and eigenvectors.
Addition: The operation of adding two matrices of the same dimension.
Subtraction: The operation of subtracting one matrix from another matrix of the same dimension.
Scalar Multiplication: The operation of multiplying a matrix by a scalar, i.e., a single number.
Matrix Multiplication: The operation of multiplying two matrices to produce a third matrix.
Transpose: The operation of interchanging the rows and columns of a matrix.
Determinant: A scalar value that can be computed from a square matrix. It is used to determine whether a matrix has an inverse or not.
Inverse: The operation of finding a matrix that, when multiplied with the original matrix, yields the identity matrix.
Rank: The rank of a matrix is the maximum number of linearly independent rows or columns.
Trace: The sum of the diagonal elements of a square matrix.
Eigenvalues and Eigenvectors: Eigenvalues and eigenvectors play an important role in linear algebra. An eigenvalue is a scalar value that, when multiplied with a corresponding eigenvector, yields the same vector multiplied by the scalar value.
Matrix Exponential: The exponential of a matrix is defined as a power series of the matrix, which results in a new matrix of the same dimension.
QR Factorization: The QR factorization of a matrix decomposes it into an orthogonal matrix and an upper triangular matrix.
Singular Value Decomposition (SVD): The SVD of a matrix decomposes it into three matrices, one of which is diagonal, and the other two are orthogonal.
Orthogonal Projection: The operation of projecting one vector onto another that is orthogonal to it.
Hermitian Matrix: A Hermitian matrix is a square matrix that is equal to its own conjugate transpose.
Positive Definite Matrix: A positive-definite matrix is a square matrix that satisfies certain properties, including all eigenvalues being positive.
Adjoint: The adjoint of a matrix is similar to the transpose, but involves the conjugate transpose of the matrix.
Kronecker Product: The Kronecker product of two matrices produces a new matrix whose entries are products of the entries of the original matrices.
Jordan Form: The Jordan form of a matrix involves transforming it into a particular block-diagonal form.
Matrix Differentiation: The process of finding derivatives of matrices with respect to some variable.
"This is often referred to as a 'two by three matrix,' a '2x3 matrix,' or a matrix of dimension 2x3."
"Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra."
"Matrix multiplication represents the composition of linear maps."
"Not all matrices are related to linear algebra. This is, in particular, the case in graph theory, of incidence matrices, and adjacency matrices."
"Square matrices, matrices with the same number of rows and columns, play a major role in matrix theory."
"Square matrices of a given dimension form a noncommutative ring, which is one of the most common examples of a noncommutative ring."
"The determinant of a square matrix is a number associated with the matrix, which is fundamental for the study of a square matrix."
"A square matrix is invertible if and only if it has a nonzero determinant."
"The eigenvalues of a square matrix are the roots of a polynomial determinant."
"In geometry, matrices are widely used for specifying and representing geometric transformations and coordinate changes. In numerical analysis, many computational problems are solved by reducing them to a matrix computation."
"Matrices are used in most areas of mathematics and most scientific fields, either directly, or through their use in geometry and numerical analysis."
"Matrix theory is the branch of mathematics that focuses on the study of matrices."
"It was initially a sub-branch of linear algebra."
"Matrix theory soon grew to include subjects related to graph theory, algebra, combinatorics, and statistics."
"Matrices represent linear maps and allow explicit computations in linear algebra."
"In graph theory, there are examples of incidence matrices and adjacency matrices that are not related to linear algebra."
"In numerical analysis, many computational problems are solved by reducing them to a matrix computation."
"The study of matrices is a large part of linear algebra, and most properties and operations of abstract linear algebra can be expressed in terms of matrices."
"Matrices are used to represent a mathematical object or a property of such an object."