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If A is an m × n matrix and A T is its transpose, then the result of matrix multiplication with these two matrices gives two square matrices: A A T is m × m and A T A is n × n. Furthermore, these products are symmetric matrices. Indeed, the matrix product A A T has entries that are the inner product of a row of A with a column of A T.
Conjugate transpose. In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an complex matrix is an matrix obtained by transposing and applying complex conjugation to each entry (the complex conjugate of being , for real numbers and ). There are several notations, such as or , [ 1] , [ 2] or (often in physics) .
Adjugate matrix. In linear algebra, the adjugate of a square matrix A is the transpose of its cofactor matrix and is denoted by adj (A). [ 1][ 2] It is also occasionally known as adjunct matrix, [ 3][ 4] or "adjoint", [ 5] though the latter term today normally refers to a different concept, the adjoint operator which for a matrix is the ...
Matrix (mathematics) An m × n matrix: the m rows are horizontal and the n columns are vertical. Each element of a matrix is often denoted by a variable with two subscripts. For example, a2,1 represents the element at the second row and first column of the matrix. In mathematics, a matrix ( pl.: matrices) is a rectangular array or table of ...
Matrix decomposition. In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems.
Rotation matrix. In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix. rotates points in the xy plane counterclockwise through an angle θ about the origin of a two-dimensional Cartesian coordinate system.
Miscellanea. v. t. e. In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that ...
Orthogonal matrix. In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors . One way to express this is where QT is the transpose of Q and I is the identity matrix . This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to ...