"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."
A matrix is a rectangular array of numbers. Linear algebra heavily relies on matrices to represent linear transformations.
Introduction to Vectors: Vectors are essential to the study of linear algebra, and their properties are fundamental to understanding matrices. Therefore, a beginner should have a clear understanding of what vectors are.
Introduction to Matrices: Matrices are collections of numbers arranged in rows and columns, used to describe systems of linear equations. Familiarity with matrices is essential for learning linear algebra.
Matrix Operations: Operations like addition, subtraction, multiplication, and division on matrices are essential in performing various linear Algebra operations.
Basic Matrix Transformations: By applying various matrix transformation techniques, such as rotation, scaling, and translation, you can design more complex graphics and animation.
Matrix Inversion: The ability to find the inverse of a matrix is essential for conducting Gaussian elimination, which is used to solve linear systems of equations.
Matrix Representations of Linear Transformations: Matrix representations of linear transformations enable standardization between linear transformations, allowing computational comparison between them.
Matrix Determinants: The determinant is a scalar quantity that measures the extent to which a square matrix modifies the space it operates on. An understanding of determinants is help in solving linear systems of equations.
Eigenvalues and Eigenvectors: Eigenvectors represent the directions along which a linear transformation stretches or compresses, and eigenvalues measure how much this compression or stretching occurs.
Diagonalization: An understanding of the diagonalization of a matrix is essential for many applications in fields such as engineering, science, and business.
Inner Products: Inner products arise naturally in the study of matrices and linear algebra. They help in finding vector orthogonal to each other and also help in finding magnitudes of vectors.
Orthogonal and Orthonormal Vectors: A set of vectors are orthogonal or orthonormal if they are all perpendicular to each other or have the same magnitudes. This is useful in many areas such as computer graphics, mathematical optimization, and signal processing.
Applications of Matrices: Matrices find applications in many areas such as engineering, physics, computer science, economics, and social sciences. Understanding the applications of matrices is helpful in identifying real-world problems that can be solved using linear algebra.
Square Matrix: A matrix with an equal number of rows and columns.
Rectangular Matrix: A matrix with a different number of rows and columns.
Zero Matrix: A matrix with all entries equal to zero.
Identity Matrix: A square matrix with diagonal entries equal to 1 and all other entries equal to zero.
Diagonal Matrix: A matrix with non-zero entries only on the diagonal.
Triangular Matrix: A matrix in which all entries below (or above) the diagonal are zero.
Symmetric Matrix: A square matrix that is equal to its transpose.
Skew-symmetric Matrix: A square matrix that is equal to the negative transpose of itself.
Orthogonal Matrix: A square matrix whose transpose is equal to its inverse.
Hermitian Matrix: A square matrix that is equal to its conjugate transpose.
Unitary Matrix: A matrix whose conjugate transpose is equal to its inverse.
Positive Definite Matrix: A symmetric matrix where all eigenvalues are positive.
Positive Semidefinite Matrix: A symmetric matrix where all eigenvalues are non-negative.
Negative Definite Matrix: A symmetric matrix where all eigenvalues are negative.
Negative Semidefinite Matrix: A symmetric matrix where all eigenvalues are non-positive.
Singular Matrix: A matrix whose determinant is equal to zero.
"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."