As a result of the first two properties above, the set of all skew-symmetric matrices of a fixed size forms a vector space.
scalars (the number of entries above the main diagonal); a symmetric matrix is determined by
scalars (the number of entries on or above the main diagonal).
Since this definition is independent of the choice of basis, skew-symmetry is a property that depends only on the linear operator
skew symmetric matrices can be used to represent cross products as matrix multiplications.
is odd, and since the underlying field is not of characteristic 2, the determinant vanishes.
Hence, all odd dimension skew symmetric matrices are singular as their determinants are always zero.
Thus the determinant of a real skew-symmetric matrix is always non-negative.
in the expansion of the determinant of a skew-symmetric matrix of order
(sequence A002370 in the OEIS) is and it is encoded in the exponential generating function The latter yields to the asymptotics (for
even) The number of positive and negative terms are approximatively a half of the total, although their difference takes larger and larger positive and negative values as
Three-by-three skew-symmetric matrices can be used to represent cross products as matrix multiplications.
Then, defining the matrix the cross product can be written as This can be immediately verified by computing both sides of the previous equation and comparing each corresponding element of the results.
One actually has i.e., the commutator of skew-symmetric three-by-three matrices can be identified with the cross-product of three-vectors.
Since the skew-symmetric three-by-three matrices are the Lie algebra of the rotation group
From the spectral theorem, for a real skew-symmetric matrix the nonzero eigenvalues are all pure imaginary and thus are of the form
Real skew-symmetric matrices are normal matrices (they commute with their adjoints) and are thus subject to the spectral theorem, which states that any real skew-symmetric matrix can be diagonalized by a unitary matrix.
However, it is possible to bring every skew-symmetric matrix to a block diagonal form by a special orthogonal transformation.
real skew-symmetric matrix can be written in the form
In the odd-dimensional case Σ always has at least one row and column of zeros.
More generally, every complex skew-symmetric matrix can be written in the form
This is an example of the Youla decomposition of a complex square matrix.
This defines a form with desirable properties for vector spaces over fields of characteristic not equal to 2, but in a vector space over a field of characteristic 2, the definition is equivalent to that of a symmetric form, as every element is its own additive inverse.
at the identity matrix; formally, the special orthogonal Lie algebra.
In this sense, then, skew-symmetric matrices can be thought of as infinitesimal rotations.
Another way of saying this is that the space of skew-symmetric matrices forms the Lie algebra
Moreover, since the exponential map of a connected compact Lie group is always surjective, it turns out that every orthogonal matrix with unit determinant can be written as the exponential of some skew-symmetric matrix.
the exponential representation for an orthogonal matrix reduces to the well-known polar form of a complex number of unit modulus.
More intrinsically (i.e., without using coordinates), skew-symmetric linear transformations on a vector space
is said to be skew-symmetrizable if there exists an invertible diagonal matrix