An $n\times n$ identity matrix and a reverse identity matrix are defined as:

$I_n=\left[\begin{array}{ccccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 1\hfill \end{array}\right]$, $I_n^R=\left[\begin{array}{ccccc}\hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \cdots \hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill ⋮\hfill & \hfill \text{...}\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill \end{array}\right]$

The transpose of a matrix $X$ is defined as element-wise swapping along the diagonal line, denoted as $X^T$, which is:

$X=\left[\begin{array}{ccccc}\hfill a_1\hfill & \hfill a_2\hfill & \hfill a_3\hfill & \hfill \cdots \hfill & \hfill a_n\hfill \\ \hfill b_1\hfill & \hfill b_2\hfill & \hfill b_3\hfill & \hfill \cdots \hfill & \hfill b_n\hfill \\ \hfill c_1\hfill & \hfill c_2\hfill & \hfill c_3\hfill & \hfill \cdots \hfill & \hfill c_n\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill m_1\hfill & \hfill m_2\hfill & \hfill m_3\hfill & \hfill \cdots \hfill & \hfill m_n\hfill \end{array}\right]$, $X^T=\left[\begin{array}{ccccc}\hfill a_1\hfill & \hfill b_1\hfill & \hfill c_1\hfill & \hfill \cdots \hfill & \hfill m_1\hfill \\ \hfill a_2\hfill & \hfill b_2\hfill & \hfill c_2\hfill & \hfill \cdots \hfill & \hfill m_2\hfill \\ \hfill a_3\hfill & \hfill b_3\hfill & \hfill c_3\hfill & \hfill \cdots \hfill & \hfill m_3\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill a_n\hfill & \hfill b_n\hfill & \hfill c_n\hfill & \hfill \cdots \hfill & \hfill m_n\hfill \end{array}\right]$

A Vandermonde matrix is an $(m+1)\times (n+1)$ matrix defined as:

$\mathrm{𝑉𝑎𝑛𝑑𝑒𝑟}(x_0,x_1,\cdots ,x_m)=\left[\begin{array}{ccccc}\hfill 1\hfill & \hfill x_0\hfill & \hfill x_0^2\hfill & \hfill \cdots \hfill & \hfill x_0^n\hfill \\ \hfill 1\hfill & \hfill x_1\hfill & \hfill x_1^2\hfill & \hfill \cdots \hfill & \hfill x_1^n\hfill \\ \hfill 1\hfill & \hfill x_2\hfill & \hfill x_2^2\hfill & \hfill \cdots \hfill & \hfill x_2^n\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill 1\hfill & \hfill x_m\hfill & \hfill x_m^2\hfill & \hfill \cdots \hfill & \hfill x_m^n\hfill \end{array}\right]$