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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.
for some skew-symmetric matrix A; more generally any orthogonal matrix Q can be written as = (+) for some skew-symmetric matrix A and some diagonal matrix E with ±1 as entries. [4] A slightly different form is also seen, [5] [6] requiring different mappings in each direction,
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...
since the matrices A and A T commute, this can be easily proven with the skew-symmetric matrix condition. This is not enough to show that 𝖘𝖔(3) is the corresponding Lie algebra for SO(3), and shall be proven separately. The level of difficulty of proof depends on how a matrix group Lie algebra is defined.
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.
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. Then, the 4D rotation matrices can be obtained from the skew-symmetric matrices A 1 and A 2 by Rodrigues' rotation formula and the Cayley formula. [9] Let A be a 4 × 4 ...
If the underlying field has characteristic not 2, alternation is equivalent to skew-symmetry. If the characteristic is 2, the skew-symmetry is implied by, but does not imply alternation. In this case every symplectic form is a symmetric form, but not vice versa. Working in a fixed basis, can be represented by a matrix.
where denotes the transpose of and is a fixed nonsingular, skew-symmetric matrix. This definition can be extended to 2 n × 2 n {\displaystyle 2n\times 2n} matrices with entries in other fields , such as the complex numbers , finite fields , p -adic numbers , and function fields .