minrankA
MinrankA is a parameter used in linear algebra and combinatorial optimization to denote the minimum possible rank of a matrix that completes or conforms to a given sparsity pattern A. In many contexts A is a binary matrix indicating where nonzero entries may occur. The notation minrankA often refers to the minimum rank, over a specified field F, among all matrices X that satisfy: X_ij ≠ 0 whenever A_ij ≠ 0 and X_ij = 0 whenever A_ij = 0. Some definitions also require diagonal entries to be nonzero or impose additional constraints.
Special case: graphs. If A encodes the adjacency pattern of a graph, then minrank over F is
Properties and complexity. The value of minrankA depends on the underlying field F; different fields can yield
Applications. MinrankA provides insights into how the structure of A constrains linear representations, with implications for
See also. Rank minimization; matrix completion; minrank of a graph; index coding.