1. What Is a Skew-Symmetric Matrix?
A skew-symmetric matrix (also called an antisymmetric matrix) is a square matrix that satisfies the condition:
\( A^T = -A \)
This means each element reflects as the negative of its counterpart across the main diagonal:
\( a_{ij} = -a_{ji} \)
So if an entry above the diagonal is 5, the corresponding entry below the diagonal must be -5.
2. Key Structural Features
For a matrix to be skew-symmetric:
- It must be square.
- Entries across the main diagonal must be negatives of each other.
- The main diagonal entries are always 0 because \( a_{ii} = -a_{ii} \Rightarrow a_{ii} = 0 \).
2.1. Example of a Skew-Symmetric Matrix
\( K = \begin{bmatrix} 0 & 3 & -4 \\ -3 & 0 & 7 \\ 4 & -7 & 0 \end{bmatrix} \)
Here:
- \( k_{12} = 3 \) and \( k_{21} = -3 \)
- \( k_{13} = -4 \) and \( k_{31} = 4 \)
- \( k_{23} = 7 \) and \( k_{32} = -7 \)
The diagonal entries are all 0, fulfilling the skew-symmetric condition.
2.2. More Examples
- \( \begin{bmatrix} 0 & 5 \\ -5 & 0 \end{bmatrix} \) — 2 × 2 skew-symmetric matrix.
- \( \begin{bmatrix} 0 & -2 & 1 \\ 2 & 0 & 7 \\ -1 & -7 & 0 \end{bmatrix} \)
- Every 1 × 1 skew-symmetric matrix must be [0].
3. How to Identify a Skew-Symmetric Matrix
To check whether a matrix is skew-symmetric, look for:
- It is a square matrix.
- The main diagonal consists of 0s.
- Every pair of entries obeys \( a_{ij} = -a_{ji} \).
If even one diagonal entry is non-zero, or one pair doesn't match this rule, the matrix is not skew-symmetric.
4. Important Observations
Skew-symmetric matrices have some interesting features:
- The matrix is fully determined by its entries above (or below) the diagonal.
- Off-diagonal elements always appear in opposite-sign pairs.
- Skew-symmetric matrices appear in vector cross products, rotations, and physics applications.
The strict structure of sign changes makes these matrices easy to recognise once you see the pattern.