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 that the transpose of the matrix is equal to its negative. In other words, for every element in the matrix:
\[ a_{ij} = -a_{ji} \]
for all \( i \) and \( j \). The diagonal elements of a skew-symmetric matrix are always zero because:
\[ a_{ii} = -a_{ii} \Rightarrow 2a_{ii} = 0 \Rightarrow a_{ii} = 0 \]
Examples of Skew-Symmetric Matrices
Example 1: A 2×2 Skew-Symmetric Matrix
\[ A = \begin{bmatrix} 0 & -3 \\ 3 & 0 \end{bmatrix} \]
Here, \( A^T = \begin{bmatrix} 0 & 3 \\ -3 & 0 \end{bmatrix} = -A \), proving that \( A \) is skew-symmetric.
Example 2: A 3×3 Skew-Symmetric Matrix
\[ B = \begin{bmatrix} 0 & -2 & 5 \\ 2 & 0 & -4 \\ -5 & 4 & 0 \end{bmatrix} \]
Checking transposition:
\[ B^T = \begin{bmatrix} 0 & 2 & -5 \\ -2 & 0 & 4 \\ 5 & -4 & 0 \end{bmatrix} = -B \]
Thus, \( B \) is a skew-symmetric matrix.
Properties of Skew-Symmetric Matrices
Property 1: Diagonal Elements are Always Zero
As shown earlier, the diagonal elements of a skew-symmetric matrix satisfy:
\[ a_{ii} = -a_{ii} \Rightarrow a_{ii} = 0 \]
Example:
\[ C = \begin{bmatrix} 0 & -4 & 2 \\ 4 & 0 & -3 \\ -2 & 3 & 0 \end{bmatrix} \]
Here, all diagonal elements are 0.
Property 2: The Sum of Two Skew-Symmetric Matrices is Skew-Symmetric
If \( A \) and \( B \) are two skew-symmetric matrices, then their sum \( A + B \) is also skew-symmetric:
\[ (A + B)^T = A^T + B^T = (-A) + (-B) = -(A + B) \]
Example:
Let
\[ A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & -2 \\ 2 & 0 \end{bmatrix} \]
Then
\[ A + B = \begin{bmatrix} 0 & -3 \\ 3 & 0 \end{bmatrix} \]
which is also skew-symmetric.
Property 3: Scalar Multiplication
If \( A \) is a skew-symmetric matrix and \( k \) is any scalar, then \( kA \) is also skew-symmetric:
\[ (kA)^T = kA^T = k(-A) = -kA \]
Example:
Let
\[ A = \begin{bmatrix} 0 & -3 \\ 3 & 0 \end{bmatrix} \]
Multiply by \( k = 2 \):
\[ 2A = \begin{bmatrix} 0 & -6 \\ 6 & 0 \end{bmatrix} \]
which is still skew-symmetric.
Property 4: Determinant of a Skew-Symmetric Matrix
The determinant of an odd-order skew-symmetric matrix is always zero. That is, if \( A \) is an \( (2n+1) \times (2n+1) \) skew-symmetric matrix, then:
\[ \det(A) = 0 \]
Example:
Consider the 3×3 skew-symmetric matrix:
\[ A = \begin{bmatrix} 0 & -2 & 5 \\ 2 & 0 & -4 \\ -5 & 4 & 0 \end{bmatrix} \]
Computing the determinant:
\[ \det(A) = 0 \]
Property 5: Eigenvalues of a Skew-Symmetric Matrix
The eigenvalues of a skew-symmetric matrix are either zero or purely imaginary numbers (i.e., of the form \( bi \) where \( b \) is a real number).
Example:
For the skew-symmetric matrix:
\[ A = \begin{bmatrix} 0 & -2 \\ 2 & 0 \end{bmatrix} \]
The eigenvalues are \( \lambda = \pm 2i \), which are purely imaginary.
Conclusion
Skew-symmetric matrices have unique properties that make them useful in various mathematical and engineering applications, particularly in physics, linear algebra, and computer graphics. Understanding their properties helps in simplifying calculations and analyzing transformations.