Strictly Upper Triangular Matrix — Definition, Formula & Examples

A strictly upper triangular matrix is a square matrix where every entry on the main diagonal and below it is zero. The only potentially nonzero entries lie above the main diagonal.

A square matrix $A = [a_{ij}]$ of size $n \times n$ is strictly upper triangular if $a_{ij} = 0$ for all $i \geq j$ , where $i$ denotes the row index and $j$ the column index.

Key Formula

a_{ij} = 0 \quad \text{for all } i \geq j

Where:

$a_{ij}$ = Entry in row i and column j of the matrix
$i$ = Row index (1 to n)
$j$ = Column index (1 to n)

How It Works

To check whether a matrix is strictly upper triangular, inspect every entry at or below the main diagonal — they must all be zero. Entries above the diagonal (where the column index exceeds the row index) can be any value. Every strictly upper triangular matrix is also an upper triangular matrix, but the reverse is not true because upper triangular matrices may have nonzero diagonal entries. A key algebraic property: every

n \times n

strictly upper triangular matrix is nilpotent, meaning

A^n = 0

. This follows because each multiplication by

A

shifts nonzero entries one position further above the diagonal until nothing remains.

Worked Example

Problem: Determine whether the following 3×3 matrix is strictly upper triangular, and if so, verify that its cube equals the zero matrix.

Step 1: Write out the matrix and identify the diagonal and below-diagonal entries.

A = \begin{bmatrix} 0 & 2 & 5 \\ 0 & 0 & 3 \\ 0 & 0 & 0 \end{bmatrix}

Step 2: Check the condition: all entries where i ≥ j are zero. The diagonal entries (1,1), (2,2), (3,3) are 0, and the below-diagonal entries (2,1), (3,1), (3,2) are also 0. The matrix is strictly upper triangular.

Step 3: Compute A² and then A³ to verify nilpotency.

A^2 = \begin{bmatrix} 0 & 0 & 6 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}, \quad A^3 = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}

Answer: The matrix is strictly upper triangular, and

A^3 = 0

, confirming it is nilpotent of index at most 3.

Why It Matters

Strictly upper triangular matrices appear naturally when decomposing matrices into diagonal and off-diagonal parts, such as in iterative methods like Gauss-Seidel for solving linear systems. Their guaranteed nilpotency makes them essential in the study of the Jordan normal form and in computing matrix exponentials.

Common Mistakes

Mistake: Confusing upper triangular with strictly upper triangular.

Correction: An upper triangular matrix can have nonzero diagonal entries. A strictly upper triangular matrix requires all diagonal entries to be zero as well.