Definition:Block Triangular Matrix

Definition

A block triangular matrix $\mathbf A$ is a square block matrix with submatrices $A_{i j}$ such that either:

$A_{i j}$ is a zero matrix for $i > j$

or:

$A_{i j}$ is a zero matrix for $i < j$.

Upper

A block upper triangular matrix $\mathbf A$ is a block triangular matrix with submatrices $A_{i j}$ such that $A_{i j}$ is a zero matrix for $i > j$.

Lower

A block lower triangular matrix $\mathbf A$ is a block triangular matrix with submatrices $A_{i j}$ such that $A_{i j}$ is a zero matrix for $i < j$.

Examples

Arbitrary Example

This $4 \times 4$ square matrix is an example of a $2 \times 2$ block upper triangular matrix wth $2 \times 2$ submatrices:

$\paren {\begin {array} {rr|rr} 1 & 2 & 5 & 4 \\ 1 & 1 & 1 & 2 \\ \hline 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & 1 \end {array} }$

which is not an upper triangular matrix.

Also see

• Results about block triangular matrices can be found here.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): block triangular matrix