Submatrices

Definition

An index set of $\upto{n}$ is a subset of $\upto{n}$. A submatrix of an $m \times n$ matrix is a matrix whose are selected according to (TODO) an index set of $\upto{m}$ and index set of $\upto{n}$; we call the first index set the row index set and the second index set the column index set. A principal submatrix is the submatrix selected when the row and column index sets are identical.

A sequential partition of $\upto{n}$ is a sequence of index sets such that all elements of a later piece of the partition are larger (in the natural order) than all elements in all previous pieces.