Definition 2.2.41. Symmetric matrix.
A symmetric matrix is a square matrix that is equal to its transpose.
Subsection 2.2 Symmetric and skew-symmetric matrices
Example 2.2.42. Symmetric matrices.
The following matrices are symmetric
\begin{equation*}
\left[ \begin{array}{rr}
0 \amp 1\\
1 \amp 3
\end{array}\right], \hspace{0.7cm}
\left[ \begin{array}{rrr}
-1 \amp 2 \amp -3 \\
2 \amp 0 \amp 1\\
-3 \amp 1 \amp 5
\end{array}\right], \hspace{0.7cm}
\left[ \begin{array}{rrrr}
1 \amp 1 \amp 4 \amp 6\\
1 \amp -2 \amp 2 \amp 5\\
4 \amp 2 \amp 4 \amp 3\\
6 \amp 5 \amp 3 \amp 1
\end{array}\right]
\end{equation*}
The following matrices are not symmetric
\begin{equation*}
\left[ \begin{array}{rr}
1 \amp 0\\
1 \amp 3
\end{array}\right], \hspace{0.7cm}
\left[ \begin{array}{rrr}
-1 \amp 2 \amp -3 \\
-2 \amp 0 \amp 1\\
-3 \amp 1 \amp 5
\end{array}\right], \hspace{0.7cm}
\left[ \begin{array}{rrrr}
1 \amp 1 \amp 4 \amp 3\\
1 \amp -2 \amp 2 \amp 7\\
3 \amp 2 \amp 4 \amp 3\\
6 \amp 1 \amp 3 \amp 1
\end{array}\right]
\end{equation*}
Checkpoint 2.2.43. Determine if a matrix is symmetric.
Determine whether the given matrix is symmetric or not
\begin{equation*}
\begin{array}{lrrrr}
\lceil \amp {-3} \amp {3} \amp {-7} \amp \rceil\\
\vert \amp {-3} \amp {8} \amp {5} \amp \vert \\
\lfloor \amp {7} \amp {-5} \amp {5} \amp \rfloor
\end{array}
\end{equation*}
- Symmetric
- Not Symmetric
\begin{equation*}
\begin{array}{lrrr}
\lceil \amp {-1} \amp {7} \amp \rceil\\
\vert \amp {7} \amp {8} \amp \vert\\
\lfloor \amp {-3} \amp {7} \amp \rfloor
\end{array}
\end{equation*}
- Symmetric
- Not Symmetric
\begin{equation*}
\begin{array}{lrrrr}
\lceil \amp {-1} \amp {7} \amp {5} \amp \rceil\\
\vert \amp {7} \amp {7} \amp {8} \amp \vert\\
\lfloor \amp {5} \amp {8} \amp {5} \amp \rfloor
\end{array}
\end{equation*}
- Symmetric
- Not Symmetric
\(\, \)
Definition 2.2.44. Skew-symmetric matrix.
A skew-symmetric matrix is a square matrix whose transpose equals its negative.
\begin{equation*}
\boldsymbol{A}^{\mathsf{T}} = - \boldsymbol{A}.
\end{equation*}
Example 2.2.45. Skew-symmetric matrices.
The following matrix is skew symmetric,
\begin{equation*}
\boldsymbol{A} = \left[\begin{array}{rrr}
1 \amp -2 \amp 3 \\
2 \amp 0 \amp 4\\
-3 \amp -4 \amp 1
\end{array}\right].
\end{equation*}
Note that
\begin{equation*}
\begin{array}{ccc}
a_{12}\amp = \amp - a_{21}\\
a_{13}\amp = \amp - a_{31}\\
a_{23}\amp = \amp - a_{32}
\end{array}
\end{equation*}
