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{{Main|Frobenius matrix}}
An '''atomic''' (upper or lower) '''triangular matrix''' is a special form of unitriangular matrix, where all of the [[off-diagonal element]]s are zero, except for the entries in a single column. Such a matrix is also called a '''Frobenius matrix''', a '''Gauss matrix''', or a '''Gauss transformation matrix'''.
=== Block triangular matrix ===
{{Main|Block matrix}}
A block triangular matrix is a [[block matrix]] (partitioned matrix) that is a triangular matrix.
====Upper block triangular====
A matrix <math>A</math> is '''upper block triangular''' if
:<math>A = \begin{bmatrix}
A_{11} & A_{12} & \cdots & A_{1k} \\
0 & A_{22} & \cdots & A_{2k} \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & A_{kk}
\end{bmatrix}</math>,
where <math>A_{ij} \in \mathbb{F}^{n_i \times n_j}</math> for all <math>i, j = 1, \ldots, k</math>.<ref name="bernstein2009">{{Cite book |last=Bernstein |first=Dennis S. |title=Matrix mathematics: theory, facts, and formulas |date= |publisher=Princeton University Press |year=2009 |isbn=978-0-691-14039-1 |edition=2 |___location=Princeton, NJ |pages=168 |language=en}}</ref>
====Lower block triangular====
A matrix <math>A</math> is '''lower block triangular''' if
:<math>A = \begin{bmatrix}
A_{11} & 0 & \cdots & 0 \\
A_{21} & A_{22} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
A_{k1} & A_{k2} & \cdots & A_{kk}
\end{bmatrix}</math>,
where <math>A_{ij} \in \mathbb{F}^{n_i \times n_j}</math> for all <math>i, j = 1, \ldots, k</math>.<ref name="bernstein2009" />
==Triangularisability{{Anchor|Triangularizability}}==
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