Handbook of Linear Algebra, 2nd Edition

Chapter 11

Partitioned Matrices

Robert Reams

SUNY-Plattsburgh

11.1 Submatrices and Block Matrices

Definitions:

Let A ∈ Fm × n. Then the row indices of A are {1, ..., m}, and the column indices of A are {1, ..., n}. Let α, β be nonempty sets of indices with α ⊆ {1, ..., m} and β ⊆ {1, ..., n}.

A submatrix A[α, β] is a matrix whose rows have indices α among the row indices of A, and whose columns have indices β among the column indices of A. A(α, β) = A[αc, βc], where αc is the complement of α.

A principal submatrix is a submatrix A[α, α], denoted more compactly as A[α].

Let the set {1, ... m} be partitioned into the subsets α1, ..., αr in the usual sense of partitioning a set (so that ${\alpha }_i\cap {\alpha }_j=\cancel{0}$, for all i ≠ j, 1 ≤ i, j ≤ r, and α1 ∪ ... ∪ αr = {1, ...