Antisymmetric Matrix (Skew-Symmetric) and Properties

An antisymmetric matrix (or Skew-Symmetric) is a special type of square matrix in linear algebra.

Antisymmetric matrices find applications in various areas of mathematics and physics. They are particularly useful in studying properties related to rotations, angular momentum, and skew-symmetric bilinear forms.

Let’s consider an \(n \times n\) matrix \(A\). It is said to be antisymmetric if it satisfies the following condition:

\[
A = -A^T
\]

where \(A^T\) represents the transpose of matrix \(A\).

In other words, the elements of an antisymmetric matrix exhibit a particular pattern. For any element \(a_{ij}\) in the matrix, its corresponding element \(a_{ji}\) is the negative of \(a_{ij}\). This can be mathematically represented as:

\[
a_{ij} = -a_{ji}
\]

for all \(i\) and \(j\).

To visualize an antisymmetric matrix, let’s consider a small example. Suppose we have a \(3 \times 3\) matrix \(A\) as follows:

\[
A = \begin{bmatrix}
0 & a & b \\
-a & 0 & c \\
-b & -c & 0
\end{bmatrix}
\]

In this case, the matrix is antisymmetric because \(a_{12} = -a_{21} = a\), \(a_{13} = -a_{31} = b\), and \(a_{23} = -a_{32} = c\).

 

Properties

1. The diagonal elements of an antisymmetric matrix are always zero.
2. The sum of an antisymmetric matrix and its transpose is the zero matrix.
3. The determinant of an antisymmetric matrix is zero if \(n\) is odd. If \(n\) is even, the determinant may be nonzero, but it follows a specific pattern.

 

Related | Symmetric Matrix and Properties