Triangular Matrices

Definition: Upper Triangular Matrix (Right Triangular Matrix)

An upper or right triangular matrix is a square matrix which has non-zero entries only on its diagonal and above it.

$$\left[\begin{array}{cccc}\ast & \ast & \cdots & \ast \\ 0& \ddots & \ddots & ⋮\\ ⋮& \ddots & \ddots & \ast \\ 0& \cdots & 0& \ast \end{array}\right]$$

Definition: Lower Triangular Matrix (Left Triangular Matrix)

A lower or left triangular matrix is a square matrix which has non-zero entries only on its diagonal and below it.

$$\left[\begin{array}{cccc}\ast & 0& \cdots & 0\\ \ast & \ddots & \ddots & ⋮\\ ⋮& \ddots & \ddots & 0\\ \ast & \cdots & \ast & \ast \end{array}\right]$$