Mastering Matrices

Matrix Basics

A matrix is a rectangular array or table of numbers, symbols, or expressions arranged in rows and columns.
Matrices represent linear maps and allow explicit computations in linear algebra.
Square matrices play a major role in matrix theory and form a noncommutative ring.
The determinant of a square matrix is a number associated with the matrix and is fundamental for its study.
Matrices are widely used in geometry for specifying and representing geometric transformations and coordinate changes.
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.
The size of a matrix is defined by the number of rows and columns it contains.
Matrices with the same number of rows and columns are called square matrices.
The entry in the i-th row and j-th column of a matrix is sometimes referred to as the i,j or (i, j) entry of the matrix, and commonly denoted by ai,j or aij.
Matrices are subject to standard operations such as addition and multiplication.
The set of all m-by-n matrices over a field or ring R is denoted by M(m,n,R).Matrices: Basic Operations and Types
Matrices can be modified using operations such as matrix addition, scalar multiplication, transposition, matrix multiplication, row operations, and submatrix.
Addition, scalar multiplication, and transposition of matrices follow familiar properties of numbers, such as commutativity and compatibility with each other.
Multiplication of matrices is defined when the number of columns of the left matrix is the same as the number of rows of the right matrix.
Matrix multiplication is not commutative, unlike the product of rational, real, or complex numbers.
Row operations include interchange, scaling, and replacement, and are used to solve linear equations and find matrix inverses.
A submatrix of a matrix is obtained by deleting any collection of rows and/or columns. The minors and cofactors of a matrix are found by computing the determinant of certain submatrices.
Matrices can be used to compactly write and work with multiple linear equations, that is, systems of linear equations.
Matrices and matrix multiplication reveal their essential features when related to linear transformations, also known as linear maps.
A square matrix is a matrix with the same number of rows and columns. Any two square matrices of the same order can be added and multiplied.
Main types of matrices include diagonal, triangular, identity, symmetric, and skew-symmetric matrices.
The identity matrix is a special kind of diagonal matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0.
Real symmetric matrices and complex Hermitian matrices have an eigenbasis.Overview of Matrices in Linear Algebra
Matrices are rectangular arrays of numbers that are used to represent linear transformations and solve systems of linear equations.
Matrix multiplication is not commutative, and it is defined by multiplying the rows of one matrix with the columns of the other.
Eigenvalues and eigenvectors are important concepts in linear algebra. Eigenvalues are the scalars that satisfy a certain equation, and eigenvectors are the vectors that get scaled by the matrix when multiplied by it.
The determinant of a matrix is a number that encodes certain properties of the matrix, such as whether it is invertible or not.
Matrix decomposition or factorization techniques are used to render matrices into a more easily accessible form. LU decomposition, Gaussian elimination, singular value decomposition, and eigendecomposition are some examples of matrix decomposition methods.
Matrices can be generalized to fields, rings, and even tensors. Abstract algebra uses matrices with entries in more general fields or rings, while linear algebra codifies properties of matrices in the notion of linear maps.
The trace of a matrix is the sum of its diagonal entries, and it is independent of the order of the factors in the product of two matrices.
Orthogonal matrices are square matrices with real entries whose columns and rows are orthogonal unit vectors. They are necessarily invertible, unitary, and normal.
Positive-definite matrices are symmetric real matrices that have a positive value for every nonzero vector x in Rn. They have all positive eigenvalues and are invertible.
Numerical linear algebra studies the effectiveness and precision of matrix algorithms. It is important to determine both the complexity of algorithms and their numerical stability.Matrices: Definition, Properties, and Applications
Matrices are arrays of numbers or mathematical objects arranged in rows and columns.
Matrices can have entries from any ring, and matrices over superrings are called supermatrices.
Linear maps between finite-dimensional vector spaces can be described by matrices, where the columns of the matrix express the image of the basis vectors of the domain space in terms of the basis vectors of the codomain space.
Matrix groups are groups of matrices with a binary operation of matrix multiplication, and they can be used to study general groups using representation theory.
Infinite matrices can be used to describe linear maps on Hilbert spaces, but they must satisfy certain constraints such as row or column finiteness.
Empty matrices are matrices with zero rows or columns, and they are useful in dealing with maps involving the zero vector space.
Matrices have numerous applications in mathematics and other sciences such as game theory, economics, text mining, and automated thesaurus compilation.
Matrices can be used to represent complex numbers, quaternions, and Clifford algebras.
Matrices are used in computer graphics to represent objects and apply image convolutions such as sharpening, blurring, and edge detection.
Matrices are used in chemistry to discuss molecular bonding and spectroscopy, in particular with the overlap matrix and the Fock matrix.
Matrices are used in graph theory to represent finite graphs and their properties such as adjacency and distance matrices.
Matrices are used in analysis and geometry to represent the Hessian matrix of a differentiable function, which encodes information about the local growth behavior of the function.