Question
What is a skew symmetric matrix
Answers
GPT-4o mini
A skew-symmetric matrix, also known as an antisymmetric matrix, is a square matrix \( A \) that satisfies the property:
\[
A^T = -A
\]
where \( A^T \) is the transpose of \( A \). This means that each element of the matrix satisfies the condition:
\[
a_{ij} = -a_{ji}
\]
for all \( i \) and \( j \). As a result, the diagonal elements of a skew-symmetric matrix must be zero (i.e., \( a_{ii} = 0 \) for all \( i \)).
A simple example of a \( 3 \times 3 \) skew-symmetric matrix is:
\[
A = \begin{pmatrix}
0 & 2 & -3 \\
-2 & 0 & 4 \\
3 & -4 & 0
\end{pmatrix}
\]
In this matrix:
- The off-diagonal elements satisfy \( a_{12} = -a_{21} \), \( a_{13} = -a_{31} \), and \( a_{23} = -a_{32} \).
- The diagonal elements are all zero.
Skew-symmetric matrices have several important properties, including:
1. The eigenvalues of a skew-symmetric matrix are either purely imaginary or zero.
2. The determinant of a skew-symmetric matrix of odd order is always zero.
\[
A^T = -A
\]
where \( A^T \) is the transpose of \( A \). This means that each element of the matrix satisfies the condition:
\[
a_{ij} = -a_{ji}
\]
for all \( i \) and \( j \). As a result, the diagonal elements of a skew-symmetric matrix must be zero (i.e., \( a_{ii} = 0 \) for all \( i \)).
A simple example of a \( 3 \times 3 \) skew-symmetric matrix is:
\[
A = \begin{pmatrix}
0 & 2 & -3 \\
-2 & 0 & 4 \\
3 & -4 & 0
\end{pmatrix}
\]
In this matrix:
- The off-diagonal elements satisfy \( a_{12} = -a_{21} \), \( a_{13} = -a_{31} \), and \( a_{23} = -a_{32} \).
- The diagonal elements are all zero.
Skew-symmetric matrices have several important properties, including:
1. The eigenvalues of a skew-symmetric matrix are either purely imaginary or zero.
2. The determinant of a skew-symmetric matrix of odd order is always zero.