What is Skew Symmetric Matrix?
A square matrix is said to be skew symmetric matrix, if its transpose is its negative. Symbolically it should satisfy the equation Z = -ZT. Also, a matrix will be a skew symmetric matrix if and only if it satisfies the condition xTAy = -yTAx for all the vectors of x and y. For all x, this is equivalent to xT Ax=0. The skew symmetric matrix is also termed as anti-symmetric matrix or anti-metric matrix.
Difference between Skew Symmetric Matrices and Symmetric Matrices
The skew symmetric matrix is similar to symmetric matrix except the following property: 1+1 not equal to 0 in skew symmetric matrix assuming that the core field doesn’t have the characteristic 2. Here one represents multiplicative identity and zero denotes additive identity.
The following equation helps to understand the difference between skew symmetric matrices and symmetric matrices better:
K = (1/2) (K+KT) + (1/2) (K-KT) = L+M, where K is an n x n matrix. L and M are two resultant matrices. In these two matrices, LT = L and so L is symmetric matrix and MT = -M and thus M is the skew symmetric matrix.
Skew Symmetric Matrix Properties
Listed below are the properties of Skew Symmetric Matrix:
1. Addition of two skew symmetric matrices will result in a skew symmetric matrix.
2. The scalar multiplication of two skew symmetric matrices will result in a skew symmetric matrix. The skew symmetric matrices are of dimension n (n-1)/2 and they form vector space.
3. The dimension m (m-1) / 2 scalars determine the skew symmetric matrix. These scalars represent the count of entries located on top of main diagonal of the matrix.
4. In a skew symmetric matrix, all the entries in main diagonal are zero. Therefore, trace value is also zero.
5. Cross products can be represented as matrix multiplications using skew symmetric matrices of dimension 3 x 3.
6. All the Eigen values are zero in value or they can be considered as purely imaginary.
Does a Skew Symmetric Matrix have an Inverse?
The skew symmetric matrix does not have an inverse, however the symmetric matrix has inverse. The symmetric matrix inverse for a diagonal matrix is calculated by replacing the diagonal elements with their reciprocal. The elements other than the diagonal elements will have the value zero in the diagonal matrix. In case of m x m matrix, the inverse of symmetric matrix is calculated with the help of determinant.
