A square matrix of the form
\[L=\begin{bmatrix} \alpha_{11} & 0 & \cdots & \cdots & 0 \\ \alpha_{21} & \alpha_{22} & 0 & \cdots & \vdots \\ \alpha_{31} & \alpha_{32} & \ddots & \ddots & \vdots \\ \vdots & \vdots & \ddots & \ddots & 0 \\ \alpha_{n1} & \alpha_{n2} & \ldots & \alpha_{n,n-1} & \alpha_{nn} \end{bmatrix}\] is called a lower triangular matrix.
Analogously a square matrix of the form
\[U = \begin{bmatrix} \alpha_{11} & \alpha_{12} & \alpha_{13} & \ldots & \alpha_{1n} \\ 0 & \alpha_{22} & \alpha_{23} & \ldots & \alpha_{2n} \\ \vdots & 0 & \ddots & \ddots & \vdots \\ \vdots & \vdots & \ddots & \ddots & \alpha_{n-1,n}\\ 0 & 0 & \cdots & 0 & \alpha_{nn} \end{bmatrix}\] is called an upper triangular matrix.
Please note that both matrices are square matrices: \(L,U\in M_{n\times n}(F)\).
