"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."
Introduction to matrices and determinants and how they related to geometry.
Introduction to matrices: Definition, types of matrices (square, rectangular, row, column), elements of a matrix.
Basic matrix operations: Addition, subtraction, scalar multiplication, matrix multiplication.
Properties of matrices: Commutativity, associativity, distributivity, identity matrix, inverse matrix.
Transpose of a matrix: Definition, properties, transpose of a product of matrices.
Determinants: Definition, properties, computation of determinants, properties of invertible matrices, Cramer's rule.
Inverse of a matrix: Definition, computation of inverse, properties of invertible matrices.
Systems of linear equations: Representing systems of linear equations using matrices, solving systems of linear equations using matrices and determinants.
Eigenvalues and eigenvectors of a matrix: Definition, computation of eigenvalues and eigenvectors, properties.
Diagonalization: Definition, computation of diagonalization, eigendecomposition, uses.
Applications of matrices and determinants: Linear transformations, computer graphics, Markov chains, statistics, optimization problems, cryptography, engineering, physics.
Square Matrix: A square matrix is a matrix that has the same number of rows and columns. It is of the form A[nx n].
Identity Matrix: An identity matrix is a square matrix where every diagonal element is equal to 1, and all other elements are zero. It is of the form I[nx n].
Zero Matrix: A zero matrix, also known as a null matrix, is a matrix in which every element is equal to zero. It is of the form O[nx n].
Diagonal Matrix: A diagonal matrix is a square matrix in which all the off-diagonal elements are zero. The diagonal elements can be any scalar value. It is of the form D[nx n].
Upper Triangular Matrix: An upper triangular matrix is a square matrix in which all the elements below the diagonal are zero. It is of the form U[nx n].
Lower Triangular Matrix: A lower triangular matrix is a square matrix in which all the elements above the diagonal are zero. It is of the form L[nx n].
Symmetric Matrix: A symmetric matrix is a square matrix that is equal to its transpose. It is of the form S[nx n].
Skew-Symmetric Matrix: A skew-symmetric matrix is a square matrix that is equal to the negative of its transpose. It is of the form A[nx n].
Orthogonal Matrix: An orthogonal matrix is a square matrix in which every row and column is orthonormal. It is of the form Q[nx n].
Hermitian Matrix: A Hermitian matrix is a complex square matrix whose transpose is equal to its complex conjugate. It is of the form H[nx n].
Normal Matrix: A normal matrix is a complex square matrix that commutes with its conjugate transpose. It is of the form N[nx n].
Positive Definite Matrices: A positive definite matrix is a symmetric matrix whose eigenvalues are all positive. It is of the form P[nx n].
Determinants: A determinant is a scalar value that can be computed from a square matrix. It can be used to find the inverse of the matrix and to solve systems of linear equations.
"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."