Matrix - Triangular

A triangular matrix is a square matrix that has all its entries either above or below the main diagonal equal to zero.

Types

There are two types of triangular matrices:

1. Upper Triangular Matrix: A matrix is called upper triangular if all the entries below the main diagonal are zero. For example, the following matrix is upper triangular:

$$\left[\begin{array}{ccc}2& 3& 1\\ 0& 5& 4\\ 0& 0& 6\end{array}\right]$$

2. Lower Triangular Matrix: A matrix is called lower triangular if all the entries above the main diagonal are zero. For example, the following matrix is lower triangular:

$$\left[\begin{array}{ccc}7& 0& 0\\ 2& 8& 0\\ 3& 4& 9\end{array}\right]$$

Properties

Determinant

The determinant of a triangular matrix is the product of its diagonal entries. For example, for the upper triangular matrix above, the determinant is:

$$\text{det}=2\times 5\times 6=60$$

Eigenvalues

The eigenvalues of a triangular matrix are the entries on its main diagonal.