Editing Block matrix (section)

==Formal definition==
Let <math>A \in \mathbb{C}^{m \times n}</math>. A '''partitioning''' of <math>A</math> is a representation of <math>A</math> in the form

:<math>A = \begin{bmatrix}
A_{11} & A_{12} & \cdots & A_{1q} \\
A_{21} & A_{22} & \cdots & A_{2q} \\
\vdots & \vdots & \ddots & \vdots \\
A_{p1} & A_{p2} & \cdots & A_{pq}
\end{bmatrix}</math>,

where <math>A_{ij} \in \mathbb{C}^{m_i \times n_j}</math> are contiguous submatrices, <math>\sum_{i=1}^{p} m_i = m</math>, and <math>\sum_{j=1}^{q} n_j = n</math>.<ref name=":2">{{Cite book |last=Stewart |first=Gilbert W. |title=Matrix algorithms. 1: Basic decompositions |date=1998 |publisher=Soc. for Industrial and Applied Mathematics |isbn=978-0-89871-414-2 |location=Philadelphia, PA |pages=18–20}}</ref> The elements <math>A_{ij}</math> of the partition are called '''blocks'''.<ref name=":2" />

By this definition, the blocks in any one column must all have the same number of columns.<ref name=":2" /> Similarly, the blocks in any one row must have the same number of rows.<ref name=":2" />

=== Partitioning methods ===
A matrix can be partitioned in many ways.<ref name=":2" /> For example, a matrix <math>A</math> is said to be '''partitioned by columns''' if it is written as

:<math>A = (a_1 \ a_2 \ \cdots \ a_n)</math>,

where <math>a_j</math> is the <math>j</math>th column of <math>A</math>.<ref name=":2" /> A matrix can also be '''partitioned by rows''':

:<math>A = \begin{bmatrix}
a_1^T \\
a_2^T \\
\vdots \\
a_m^T
\end{bmatrix}</math>,

where <math>a_i^T</math> is the <math>i</math>th row of <math>A</math>.<ref name=":2" />

=== Common partitions ===
Often,<ref name=":2" /> we encounter the 2x2 partition

:<math>A = \begin{bmatrix}
A_{11} & A_{12} \\
A_{21} & A_{22}
\end{bmatrix}</math>,<ref name=":2" />

particularly in the form where <math>A_{11}</math> is a scalar:

:<math>A = \begin{bmatrix}
a_{11} & a_{12}^T \\
a_{21} & A_{22}
\end{bmatrix}</math>.<ref name=":2" />