Lower Triangular Matrix

A square matrix is called a lower triangular matrix if all the elements above the main diagonal are zero: $$ \forall \ i < j \ \ \ \ a_{ij } = 0 $$

A Practical Example

Here’s an example of a 3×3 lower triangular matrix, with three rows and three columns.

All the elements above the main diagonal are zero.

Note: The other elements of the matrix don’t necessarily have to be nonzero—they can be zero as well.

If the main diagonal consists entirely of zeros, the lower triangular matrix is called a strictly lower triangular matrix.

The set of lower triangular matrices with real coefficients of order \( n \) is denoted as \( T_{R} \).

Note: Triangular matrices reduce computational complexity because data is confined to just one part of the matrix. This means fewer calculations and less memory usage.

Why Use a Lower Triangular Matrix?

The elements of a lower triangular matrix occupy only half the space of a full square matrix. This makes computations more efficient.

Additionally, the determinant of a triangular matrix is simply the product of the elements on the main diagonal, as all other terms contribute zero.

And so on.