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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.
In numerical linear algebra, a Jacobi rotation is a rotation, Q kℓ, of a 2-dimensional linear subspace of an n-dimensional inner product space, chosen to zero a symmetric pair of off-diagonal entries of an n×n real symmetric matrix, A, when applied as a similarity transformation:
Define the n × n matrix A by = (,). The matrix A is a symmetric matrix exactly due to symmetry of the bilinear form. If we let the n×1 matrix x represent the vector v with respect to this basis, and similarly let the n×1 matrix y represent the vector w, then (,) is given by :
If the characteristic of the field is 2, then a skew-symmetric matrix is the same thing as a symmetric matrix. The sum of two skew-symmetric matrices is skew-symmetric. A scalar multiple of a skew-symmetric matrix is skew-symmetric. The elements on the diagonal of a skew-symmetric matrix are zero, and therefore its trace equals zero.
matrix is symmetric matrix.; matrix is persymmetric matrix, i.e. it is symmetric with respect to the northeast-to-southwest diagonal too.; Every one row and column of matrix consists all n elements of given vector without repetition.
In linear algebra, a skew-Hamiltonian matrix is a specific type of matrix that corresponds to a skew-symmetric bilinear form on a symplectic vector space. Let be a vector space equipped with a symplectic form, denoted by Ω. A symplectic vector space must necessarily be of even dimension.
In mathematics and multivariate statistics, the centering matrix [1] is a symmetric and idempotent matrix, which when multiplied with a vector has the same effect as subtracting the mean of the components of the vector from every component of that vector.
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