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The elements on the diagonal of a skew-symmetric matrix are zero, and therefore its trace equals zero. If is a real skew-symmetric matrix and is a real eigenvalue, then =, i.e. the nonzero eigenvalues of a skew-symmetric matrix are non-real. If is a real skew-symmetric matrix, then + is invertible, where is the identity matrix.
The matrices in the Lie algebra are not themselves rotations; the skew-symmetric matrices are derivatives, proportional differences of rotations. An actual "differential rotation", or infinitesimal rotation matrix has the form +, where dθ is vanishingly small and A ∈ so(n), for instance with A = L x,
Anti-Hermitian matrix: Synonym for skew-Hermitian matrix. Anti-symmetric matrix: Synonym for skew-symmetric matrix. Arrowhead matrix: A square matrix containing zeros in all entries except for the first row, first column, and main diagonal. Band matrix: A square matrix whose non-zero entries are confined to a diagonal band. Bidiagonal matrix
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.
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.
A 2×2 real and symmetric matrix representing a stretching and shearing of the plane. The eigenvectors of the matrix (red lines) are the two special directions such that every point on them will just slide on them. The example here, based on the Mona Lisa, provides a simple illustration. Each point on the painting can be represented as a vector ...
The abstract analog of a symplectic matrix is a symplectic transformation of a symplectic vector space. Briefly, a symplectic vector space (,) is a -dimensional vector space equipped with a nondegenerate, skew-symmetric bilinear form called the symplectic form.
The fact that the Pauli matrices, along with the identity matrix I, form an orthogonal basis for the Hilbert space of all 2 × 2 complex matrices , over , means that we can express any 2 × 2 complex matrix M as = + where c is a complex number, and a is a 3-component, complex vector.