Block matrix

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a block matrix or a partitioned matrix is a <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> that is interpreted as having been broken into sections called blocks or submatrices.
Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or <a href="/facts/Partition_of_a_set/VeP9g8CA">partition</a> it, into a collection of smaller matrices. For example, the 3x4 matrix presented below is divided by horizontal and vertical lines into four blocks: the top-left 2x3 block, the top-right 2x1 block, the bottom-left 1x3 block, and the bottom-right 1x1 block.

[
          
            
              
                
                  
                    a
                    
                      11
                    
                  
                
                
                  
                    a
                    
                      12
                    
                  
                
                
                  
                    a
                    
                      13
                    
                  
                
                
                  
                    b
                    
                      1
                    
                  
                
              
              
                
                  
                    a
                    
                      21
                    
                  
                
                
                  
                    a
                    
                      22
                    
                  
                
                
                  
                    a
                    
                      23
                    
                  
                
                
                  
                    b
                    
                      2
                    
                  
                
              
              
                
                  
                    c
                    
                      1
                    
                  
                
                
                  
                    c
                    
                      2
                    
                  
                
                
                  
                    c
                    
                      3
                    
                  
                
                
                  d
                
              
            
          
          ]
        
      
    
    {\displaystyle \left[{\begin{array}{ccc|c}a_{11}&a_{12}&a_{13}&b_{1}\\a_{21}&a_{22}&a_{23}&b_{2}\\\hline c_{1}&c_{2}&c_{3}&d\end{array}}\right]}

Any matrix may be interpreted as a block matrix in one or more ways, with each interpretation defined by how its rows and columns are partitioned.
This notion can be made more precise for an 
 
 
 
 n
 
 
 {\displaystyle n}
 
 by 
 
 
 
 m
 
 
 {\displaystyle m}
 
 matrix 
 
 
 
 M
 
 
 {\displaystyle M}
 
 by partitioning 
 
 
 
 n
 
 
 {\displaystyle n}
 
 into a collection 
 
 
 
 
 rowgroups
 
 
 
 {\displaystyle {\text{rowgroups}}}
 
, and then partitioning 
 
 
 
 m
 
 
 {\displaystyle m}
 
 into a collection 
 
 
 
 
 colgroups
 
 
 
 {\displaystyle {\text{colgroups}}}
 
. The original matrix is then considered as the "total" of these groups, in the sense that the 
 
 
 
 (
 i
 ,
 j
 )
 
 
 {\displaystyle (i,j)}
 
 entry of the original matrix corresponds in a <a href="/facts/Bijection/j4gTuTmW">1-to-1</a> way with some 
 
 
 
 (
 s
 ,
 t
 )
 
 
 {\displaystyle (s,t)}
 
 <a href="/facts/Offset_(computer_science)/Qv0N3nND">offset</a> entry of some 
 
 
 
 (
 x
 ,
 y
 )
 
 
 {\displaystyle (x,y)}
 
, where 
 
 
 
 x
 ∈
 
 rowgroups
 
 
 
 {\displaystyle x\in {\text{rowgroups}}}
 
 and 
 
 
 
 y
 ∈
 
 colgroups
 
 
 
 {\displaystyle y\in {\text{colgroups}}}
 
.
Block matrix algebra arises in general from <a href="/facts/Biproduct/HSruLv6e">biproducts</a> in <a href="/facts/Category_(mathematics)/7WzxUThf">categories</a> of matrices.

Block matrix open-in-new

Block matrix