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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.
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.
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,
A matrix with constant skew-diagonals; also an upside down Toeplitz matrix. A square Hankel matrix is symmetric. Hermitian matrix: A square matrix which is equal to its conjugate transpose, A = A *. Hessenberg matrix: An "almost" triangular matrix, for example, an upper Hessenberg matrix has zero entries below the first subdiagonal. Hollow matrix
Skew-Hermitian matrices can be understood as the complex versions of real skew-symmetric matrices, or as the matrix analogue of the purely imaginary numbers. [2] The set of all skew-Hermitian n × n {\displaystyle n\times n} matrices forms the u ( n ) {\displaystyle u(n)} Lie algebra , which corresponds to the Lie group U( n ) .
For example, the following 3×3 matrix is symmetric: [] Every square diagonal matrix is symmetric, since all off-diagonal entries are zero. Similarly, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative.
Let A be a 4 × 4 skew-symmetric matrix. The skew-symmetric matrix A can be uniquely decomposed as = + into two skew-symmetric matrices A 1 and A 2 satisfying the properties A 1 A 2 = 0, A 1 3 = −A 1 and A 2 3 = −A 2, where ∓θ 1 i and ∓θ 2 i are the eigenvalues of A.
While a rotation matrix is an orthogonal matrix = representing an element of () (the special orthogonal group), the differential of a rotation is a skew-symmetric matrix = in the tangent space (the special orthogonal Lie algebra), which is not itself a rotation matrix.