linear algebra, higher linear algebra
(…)
If we view a matrix with entries in some set with rows labelled by elements in the index set and rows labelled by elements in as a function then a partitioned matrix (also called a block matrix) is a matrix together with partitions of the sets of row labels and column labels.
For partitions , , there are corresponding submatrices, cells or blocks which are matrices defined by as long as . Conversely, is determined by for all and . Thus, one views a partitioned matrix as such a labelled collections of blocks.
One often uses some ordering on and and partitions which respect the ordering.
Gauss elimination procedure, row operations, Gauss decomposition, quasideterminants, computation of inverse matrix, make sense and are performed alike the classical case, but with the order of multiplicating blocks essential. However, the determinant does not make sense or does not behave well even when it is still accidentaly defined.
Heredetary property of quasideterminants: quasideterminant at of block quasideterminant at where , equals the quasideterminant of the original matrix at . In particular, inverse of a block matrix has entries of entries which are entries of inverse of the original matrix. Consequently, linear equations may be solved by block version of Gaussian elimination and the result reinterpreted in terms of original matrix labels.
See also
Last revised on May 21, 2024 at 16:31:12. See the history of this page for a list of all contributions to it.