Names of Matrices and their Meaning
1. Square matrices
Number of rows \(=\) number of columns. For example,
\[
P=\begin{bmatrix}E_1&E_2&E_3\\E_1'&E_2'&E_3'\\E_1''&E_2''&E_3''\end{bmatrix}_{3\times3},\qquad m=n.
\]
Transpose \(A^T\) Let
\[
A=\begin{bmatrix}E_1&E_2&E_3\\E_4&E_5&E_6\\E_7&E_8&E_9\end{bmatrix}\;\Rightarrow\;
A^T=\begin{bmatrix}E_1&E_4&E_7\\E_2&E_5&E_8\\E_3&E_6&E_9\end{bmatrix}.
\]
We write \(A=[a_{ij}]_{m\times n}\); for example \(a_{11}=E_1\), \(a_{12}=E_2\), \(a_{31}=E_7\), etc.
(a) \((A^T)^T=A\); (b) \((AB)^T=B^TA^T\); (c) \((A+B)^T=A^T+B^T\).
2. Symmetric matrix: \(A^T=A\)
\[
A=\begin{bmatrix}20&120&200\\120&10&150\\200&150&30\end{bmatrix},\qquad
A^T=\begin{bmatrix}20&120&200\\120&10&150\\200&150&30\end{bmatrix}=A.
\]
\(\therefore A\) is a symmetric matrix.
3. Skew-symmetric matrix: \(A^T=-A\)
\[
B=\begin{bmatrix}0&1&-3\\-1&0&-2\\3&2&0\end{bmatrix},\qquad
B^T=\begin{bmatrix}0&-1&3\\1&0&2\\-3&-2&0\end{bmatrix}=-\begin{bmatrix}0&1&-3\\-1&0&-2\\3&2&0\end{bmatrix}.
\]
Since \(B^T=-B\), \(B\) is skew-symmetric. The diagonal elements of a skew-symmetric matrix are always zero.
Proof If \(A^T=-A\), then \(a_{ji}+a_{ij}=0\). For \(i=j\), \(2a_{ii}=0\Rightarrow a_{ii}=0\).
4. Orthogonal matrix: \(A^TA=I A^T=A^-1</h3>
\)
The sum of the squares of the elements of any row equals \(1\):
\[
\text{if }\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}\text{ is orthogonal}\;\Rightarrow\;
a^2+b^2+c^2=1,\quad d^2+e^2+f^2=1,\quad g^2+h^2+i^2=1.
\]
Symmetric and skew-symmetric parts Any square matrix can be written as
\[
A=\frac12(A+A^T)+\frac12(A-A^T).
\]
With \(B=\frac12(A+A^T)\) and \(C=\frac12(A-A^T)\), \(A=B+C\), where \(B^T=B\) is symmetric and
\(C^T=\frac12(A^T-A)=-C\) is skew-symmetric.
Example 4.2 If \(P=\begin{bmatrix}a\\b\end{bmatrix}\begin{bmatrix}a&-2b\end{bmatrix}\), what is its symmetric part?
Solution. \(P=\begin{bmatrix}a\\b\end{bmatrix}\begin{bmatrix}a&-2b\end{bmatrix}=\begin{bmatrix}a^2&-2ab\\ba&-2b^2\end{bmatrix}\). The symmetric part is
\[
B=\frac12(P+P^T)=\frac12\left\{\begin{bmatrix}a^2&-2ab\\ba&-2b^2\end{bmatrix}+\begin{bmatrix}a^2&ab\\-2ab&-2b^2\end{bmatrix}\right\}
=\frac12\begin{bmatrix}2a^2&-ab\\-ab&-4b^2\end{bmatrix}
=\begin{bmatrix}a^2&-ab/2\\-ab/2&-2b^2\end{bmatrix}.
\]
5. Hermitian matrix: \(A^\dagger=A\)
For example, \(A=\begin{bmatrix}4&1-3\ii\\1+3\ii&7\end{bmatrix}\) has
\(A^T=\begin{bmatrix}4&1+3\ii\\1-3\ii&7\end{bmatrix}\); complex conjugation gives
\(A^\dagger=\begin{bmatrix}4&1-3\ii\\1+3\ii&7\end{bmatrix}=A\), so \(A\) is Hermitian.
6. Skew-Hermitian matrix: \(A^\dagger=-A\)
For example, \(A=\begin{bmatrix}3\ii&2+\ii\\-2+\ii&-\ii\end{bmatrix}\):
\[
A^\dagger=\begin{bmatrix}-3\ii&-2-\ii\\2-\ii&\ii\end{bmatrix}=-\begin{bmatrix}3\ii&2+\ii\\-2+\ii&-\ii\end{bmatrix},\qquad A^\dagger=-A.
\]
\(\therefore A\) is a skew-Hermitian matrix.
7. Unitary matrix: \(A^\dagger A=I\) or \(A^=A^-1</h3>
\)
8. Triangular matrix
If all elements below the principal diagonal are zero, as in \(\begin{bmatrix}a&b\\0&c\end{bmatrix}\), the matrix
is upper triangular; if all elements above it are zero, it is lower triangular.
9. Diagonal matrix
For example, \(A=\begin{bmatrix}1&0&0\\0&0&0\\0&0&0\end{bmatrix}\): all off-diagonal elements are zero, and
at least one diagonal element is non-zero.
10. Unit matrix
For example, \(A=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\): \(1\)s on the principal diagonal, zeros
elsewhere. It is also called the identity matrix.
11. Rotation matrices
Rotation matrix in 2D
\[
A=\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix},\]
\[\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}\begin{bmatrix}x_1\\y_1\end{bmatrix}=\begin{bmatrix}x_1'\\y_1'\end{bmatrix}.
\]
Rotation matrix in 3D \(+\theta\) for the anticlockwise direction; \(-\theta\) for
the clockwise direction.
\begin{gather*}
A_x(\pm\theta)=\begin{bmatrix}1&0&0\\0&\cos\theta&\mp\sin\theta\\0&\pm\sin\theta&\cos\theta\end{bmatrix},\\
A_y(\pm\theta)=\begin{bmatrix}\cos\theta&0&\pm\sin\theta\\0&1&0\\ \mp\sin\theta&0&\cos\theta\end{bmatrix},\\
A_z(\pm\theta)=\begin{bmatrix}\cos\theta&\mp\sin\theta&0\\ \pm\sin\theta&\cos\theta&0\\0&0&1\end{bmatrix}.
\end{gather*}
