\begin{definition}[Definition:Diagonal Matrix]
Let $\mathbf A = \begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & \cdots & a_{nn} \\
\end{bmatrix}$ be a square matrix of order $n$.
Then $\mathbf A$ is a '''diagonal matrix''' {{iff}} all elements of $\mathbf A$ are zero except for possibly its diagonal elements.
Thus $\mathbf A = \begin{bmatrix}
a_{11} & 0 & \cdots & 0 \\
0 & a_{22} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & a_{nn} \\
\end{bmatrix}$.
It follows by the definition of triangular matrix that a diagonal matrix is both an upper triangular matrix and a lower triangular matrix.
\end{definition}