Editing Triangular matrix (section)

==Special forms==
=== Unitriangular matrix ===

If the entries on the [[main diagonal]] of a (lower or upper) triangular matrix are all 1, the matrix is called (lower or upper) '''unitriangular'''.

Other names used for these matrices are '''unit''' (lower or upper) '''triangular''', or very rarely '''normed''' (lower or upper) '''triangular'''. However, a ''unit'' triangular matrix is not the same as '''the''' ''[[identity matrix|unit matrix]]'', and a ''normed'' triangular matrix has nothing to do with the notion of [[matrix norm]].

All finite unitriangular matrices are [[unipotent]].

=== Strictly triangular matrix ===

If all of the entries on the main diagonal of a (lower or upper) triangular matrix are also 0, the matrix is called '''strictly''' (lower or upper) '''triangular'''.

All finite strictly triangular matrices are [[nilpotent matrix|nilpotent]] of index at most ''n'' as a consequence of the [[Cayley–Hamilton theorem|Cayley-Hamilton theorem]].

=== Atomic triangular matrix ===
{{Main|Frobenius matrix}}
An '''atomic''' (lower or upper) '''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 |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" />