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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) .
Symmetric matrix. Symmetry of a 5×5 matrix. In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal.
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 ...
Vectorization (mathematics) In mathematics, especially in linear algebra and matrix theory, the vectorization of a matrix is a linear transformation which converts the matrix into a vector. Specifically, the vectorization of a m × n matrix A, denoted vec (A), is the mn × 1 column vector obtained by stacking the columns of the matrix A on top ...
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.
Symplectic matrix. In mathematics, a symplectic matrix is a matrix with real entries that satisfies the condition. (1) where denotes the transpose of and is a fixed nonsingular, skew-symmetric matrix. This definition can be extended to matrices with entries in other fields, such as the complex numbers, finite fields, p -adic numbers, and ...
Conjugate gradient, assuming exact arithmetic, converges in at most n steps, where n is the size of the matrix of the system (here n = 2). In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-semidefinite.
Laplacian matrix. In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph. Named after Pierre-Simon Laplace, the graph Laplacian matrix can be viewed as a matrix form of the negative discrete Laplace ...