Editing Block matrix (section)

===Transpose===
Let

:<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}^{k_i \times \ell_j}</math>. (This matrix <math>A</math> will be reused in {{section link||Addition}} and {{section link||Multiplication}}.) Then its transpose is

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

and the same equation holds with the transpose replaced by the conjugate transpose.<ref name=":2" />

====Block transpose====
A special form of matrix [[transpose]] can also be defined for block matrices, where individual blocks are reordered but not transposed. Let <math>A=(B_{ij})</math> be a <math>k \times l</math> block matrix with <math>m \times n</math> blocks <math>B_{ij}</math>, the block transpose of <math>A</math> is the <math>l \times k</math> block matrix <math>A^\mathcal{B}</math> with <math>m \times n</math> blocks <math>\left(A^\mathcal{B}\right)_{ij} = B_{ji}</math>.<ref>{{cite thesis |last=Mackey |first=D. Steven |date=2006 |title=Structured linearizations for matrix polynomials |publisher=University of Manchester |issn=1749-9097 |oclc=930686781 |url=http://eprints.maths.manchester.ac.uk/314/1/mackey06.pdf}}</ref> As with the conventional trace operator, the block transpose is a [[linear mapping]] such that <math>(A + C)^\mathcal{B} = A^\mathcal{B} + C^\mathcal{B} </math>.<ref name=":1" /> However, in general the property <math>(A C)^\mathcal{B} = C^\mathcal{B} A^\mathcal{B} </math> does not hold unless the blocks of <math>A</math> and <math>C</math> commute.