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Gramian matrix: The symmetric matrix of the pairwise inner products of a set of vectors in an inner product space: Hessian matrix: The square matrix of second partial derivatives of a function of several variables: Householder matrix: The matrix of a reflection with respect to a hyperplane passing through the origin: Jacobian matrix
Any square matrix can uniquely be written as sum of a symmetric and a skew-symmetric matrix. This decomposition is known as the Toeplitz decomposition. Let Mat n {\displaystyle {\mbox{Mat}}_{n}} denote the space of n × n {\displaystyle n\times n} matrices.
Matrix types (special types like bidiagonal/tridiagonal are not listed): Real – general (nonsymmetric) real; Complex – general (nonsymmetric) complex; SPD – symmetric positive definite (real) HPD – Hermitian positive definite (complex) SY – symmetric (real) HE – Hermitian (complex) BND – band; Operations: TF – triangular ...
A Jacobi operator, also known as Jacobi matrix, is a symmetric linear operator acting on sequences which is given by an infinite tridiagonal matrix. It is commonly used to specify systems of orthonormal polynomials over a finite, positive Borel measure. This operator is named after Carl Gustav Jacob Jacobi.
For a symmetric matrix A, the vector vec(A) contains more information than is strictly necessary, since the matrix is completely determined by the symmetry together with the lower triangular portion, that is, the n(n + 1)/2 entries on and below the main diagonal. For such matrices, the half-vectorization is sometimes more useful than the ...
In case of a symmetric matrix it is the largest absolute value of its eigenvectors and thus equal to its spectral radius. Condition number The condition number of a nonsingular matrix is defined as = ‖ ‖ ‖ ‖. In case of a symmetric matrix it is the absolute value of the quotient of the largest and smallest eigenvalue.
In matrix theory and combinatorics, a Pascal matrix is a matrix (possibly infinite) containing the binomial coefficients as its elements. It is thus an encoding of Pascal's triangle in matrix form. There are three natural ways to achieve this: as a lower-triangular matrix, an upper-triangular matrix, or a symmetric matrix. For example, the 5 × ...
Conversely, let Q be any orthogonal matrix which does not have −1 as an eigenvalue; then = (+) is a skew-symmetric matrix. (See also: Involution.) The condition on Q automatically excludes matrices with determinant −1, but also excludes certain special orthogonal matrices.