Matrix is a tool to deal with the linear algebra. Vector is usually used to describe most of the physical quantities those have both the direction and amplitude. The vector operations are usually very complicated in the vector algebra. A vector is a simple matrix. I try to obtain the matrix form of vector operation in order to simplify and understand the vector algebra.
In the plasma physics, as the reflective index \(\mathbf n=\frac{c\mathbf k}{\omega}\) and the electrical field vector \(\mathbf E\) are basic qualities, I use them to show the operation of two vector.
Suppose the wave vector and electrical field vector are
\[ \mathbf n=\mathbf i n_x + \mathbf j n_y +\mathbf k n_z \]
\[\mathbf E=\mathbf i E_x + \mathbf j E_y +\mathbf k E_z \]
This is their algebra form, we change it into the matrix form
\[\mathbf n=\begin{bmatrix}
n_x \\
n_y \\
n_z \\
\end{bmatrix} \,\,\,
\mathbf E=\begin{bmatrix}
E_x \\
E_y \\
E_z \\
\end{bmatrix}
\]
There are numerous ways to multiply tow vectors. The dot product takes in two vectors and returns a scalar, while the cross product returns a scalar, while the cross product returns a peseodovector. Both of these have various significant geometric interpretations and are widely used in physics and engineering.
Their dot product is
\[\mathbf n \cdot \mathbf E=n_x E_x+n_y E_y+n_z E_z=\begin{bmatrix}
n_x & n_y & n_z
\end{bmatrix}
\begin{bmatrix}
E_x \\
E_y \\
E_z \\
\end{bmatrix}
\]
Their cross product is
\[\mathbf n \times \mathbf E=\begin{bmatrix}\mathbf i & \mathbf j & \mathbf k \\
n_x & n_y & n_z \\
E_x & E_y & E_z
\end{bmatrix}=\mathbf i (n_y E_z-n_z E_y)+\mathbf j (n_z E_x-n_x E_z)+\mathbf k (n_x E_y-n_y E_x)\]
\[=\begin{bmatrix}
0 & -n_z & n_y \\
n_z & 0 & -n_x \\
-n_y & n_x & 0
\end{bmatrix}
\begin{bmatrix}
E_x \\
E_y \\
E_z \\
\end{bmatrix}
\]
The curl of the curl of the electric field is present in the Helmholtz equation. It is easy to obtain this term with the help of the previous matrix form.
\[\mathbf n \times (\mathbf n \times \mathbf E)=
\begin{bmatrix}
0 & -n_z & n_y \\
n_z & 0 & -n_x \\
-n_y & n_x & 0
\end{bmatrix}
\begin{bmatrix}
0 & -n_z & n_y \\
n_z & 0 & -n_x \\
-n_y & n_x & 0
\end{bmatrix}
\begin{bmatrix}
E_x \\
E_y \\
E_z \\
\end{bmatrix}\]
\[=
\begin{bmatrix}
-(n_y^2+n_z^2) & n_x n_y & n_x n_z \\
n_y n_x & -(n_x^2+n_z^2) & n_y n_z \\
n_z n_x & n_z n_y & -(n_x^2+n_y^2)
\end{bmatrix}
\begin{bmatrix}
E_x \\
E_y \\
E_z \\
\end{bmatrix}
\]
In the electromagnetic, the bac-cab vector identities is often used to deal with this term,
\[\mathbf n \times (\mathbf n \times \mathbf E)=\mathbf n (\mathbf n \cdot \mathbf E)-\mathbf E (\mathbf n \cdot \mathbf n)\]
\[=\begin{bmatrix}
n_x \\
n_y \\
n_z \\
\end{bmatrix}
\begin{bmatrix}
n_x & n_y & n_z
\end{bmatrix}
\begin{bmatrix}
E_x \\
E_y \\
E_z \\
\end{bmatrix}-
\begin{bmatrix}
E_x \\
E_y \\
E_z \\
\end{bmatrix}(\begin{bmatrix}
n_x & n_y & n_z
\end{bmatrix}
\begin{bmatrix}
n_x \\
n_y \\
n_z \\
\end{bmatrix})
\]
\[=\begin{bmatrix}
n_x^2 & n_x n_y & n_x n_z \\
n_y n_x & n_y^2 & n_y n_z \\
n_z n_x & n_z n_y & n_z^2
\end{bmatrix}
\begin{bmatrix}
E_x \\
E_y \\
E_z \\
\end{bmatrix}
-(n_x^2+n_y^2+n_z^2)\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}\begin{bmatrix}
E_x \\
E_y \\
E_z \\
\end{bmatrix}
\]
The first term is called the dyadic product. It takes in two vectors and returns a second order tensor. A dyadic can be used to contain physical or geometric information. However there is no direct way of geometrically interpreting it. The second term includes the identity matrix. The cross of cross of the electric field is expressed as by using the matrix notation,
\[\mathbf n \times (\mathbf n \times \mathbf E)=(\mathbf n \mathbf n-n \mathbf I ) \mathbf E\]