What is a strictly triangular matrix?

A strictly triangular matrix is a special type of square matrix where all the elements above or below the main diagonal are zero. This means that in a strictly upper triangular matrix, all entries below the diagonal are zero, while in a strictly lower triangular matrix, all entries above the diagonal are zero.

Characteristics of Strictly Triangular Matrices

• Square Shape: The matrix must be square, meaning it has the same number of rows and columns.
• Zero Elements: All elements either above or below the main diagonal are zero.
• Diagonal Elements: The diagonal elements can be any value, including zero.

Examples

Here are examples of both types:

Strictly Upper Triangular Matrix

| 1 2 3 |
| 0 4 5 |
| 0 0 6 |

Strictly Lower Triangular Matrix

| 7 0 0 |
| 8 9 0 |
| 10 11 12 |

Applications

Strictly triangular matrices are useful in various areas of mathematics, particularly in solving systems of linear equations and in matrix factorization methods.