Submatrices

Submatrices are matrices formed by selecting some subset of rows and some subset of columns from a given matrix. If A is an m-by-n matrix, then for any I ⊆ {1, ..., m} and J ⊆ {1, ..., n}, the matrix A[I,J] is the submatrix obtained by taking the rows in I and columns in J, in their original order. The size of A[I,J] is |I| by |J|. If I and J consist of consecutive indices, the resulting submatrix is called a contiguous or block submatrix.

Principal submatrices are submatrices obtained by choosing I = J; that is, A[S,S] for some S ⊆ {1, ..., m} with |S| ≤ min(m,n).

When I and J have equal cardinalities, the determinant of A[I,J] is called a minor of A; minors arising from principal submatrices are principal minors.

Block structure: Any matrix can be partitioned into submatrices (blocks) corresponding to a partition of the row and column sets, and block matrices are composed of these submatrices.

Applications and notes: Submatrices are used to study rank, linear independence, determinants, and the Schur complement, and it is common to examine minors to determine rank. In linear system analysis and algorithms, submatrices appear when restricting a matrix to certain rows or columns.

Examples: A = [[a, b, c], [d, e, f], [g, h, i]]. The submatrix with rows {1,3} and columns {2,3} is [[b, c], [h, i]]. A contiguous submatrix formed by rows 1–2 and columns 2–3 is [[b, c], [e, f]].