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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.
A skew-symmetric graph may equivalently be defined as the double covering graph of a polar graph or switch graph, [1] which is an undirected graph in which the edges incident to each vertex are partitioned into two subsets. Each vertex of the polar graph corresponds to two vertices of the skew-symmetric graph, and each edge of the polar graph ...
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. In linear algebra, a real symmetric matrix represents a self-adjoint operator over a real inner product space.
Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative. In linear algebra, a real symmetric matrix represents a self-adjoint operator [1] represented in an orthonormal basis over a real inner product space.
[nb 2] The Lie bracket of two elements of () is, as for the Lie algebra of every matrix group, given by the matrix commutator, [A 1, A 2] = A 1 A 2 − A 2 A 1, which is again a skew-symmetric matrix.
For example, if A is a 3-by-0 matrix and B is a 0-by-3 matrix, then AB is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix. There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them.
Conveniently, I − A is invertible whenever A is skew-symmetric; thus we can recover the original matrix using the Cayley transform, (+) (), which maps any skew-symmetric matrix A to a rotation matrix. In fact, aside from the noted exceptions, we can produce any rotation matrix in this way.
In general, the angular velocity in an n-dimensional space is the time derivative of the angular displacement tensor, which is a second rank skew-symmetric tensor.. This tensor Ω will have n(n−1)/2 independent components, which is the dimension of the Lie algebra of the Lie group of rotations of an n-dimensional inner product space.