Subrank

Subrank is a term used in mathematics and related disciplines to describe the size or capacity of a substructure within a larger object, with the exact definition depending on the context. In many areas, subrank serves as a way to quantify how large a subcomponent that preserves certain properties can be.

In linear algebra, subrank is sometimes used to refer to the largest order k for which a given matrix contains a k by k submatrix with full rank (i.e., a nonzero determinant) over the underlying field. In this interpretation, the subrank coincides with the standard notion of matrix rank: the maximum number of linearly independent rows or columns. As a result, subrank is often not distinguished from rank in modern linear algebra texts.

Beyond matrices, the term appears in coding theory, information theory, and related fields to denote the maximal dimension or size of a substructure that maintains a specified property. For example, it can describe the largest dimension of a subspace or subcode that satisfies certain constraints. In matroid theory, subrank can be used to describe the largest independent subset present within a given ground set, again tying into the broader idea of rank or dimension.

Because subrank has varied usage across disciplines, the precise definition is field-dependent and may differ from one context to another. When encountering the term, it is important to check the specific field’s definitions and conventions to determine what is meant by subrank in that setting. See also rank, submatrix, and minors.