In mathematics, a hollow matrix may refer to one of several related classes of matrix: a sparse matrix; a matrix with a large block of zeroes; or a matrix with diagonal entries all zero.

Definitions

Sparse

A hollow matrix may be one with "few" non-zero entries: that is, a sparse matrix.

Block of zeroes

A hollow matrix may be a square n × n matrix with an r × s block of zeroes where r + s > n .

Diagonal entries all zero

A hollow matrix may be a square matrix whose diagonal elements are all equal to zero. That is, an n × n matrix A = (aij) is hollow if aij = 0 whenever i = j (i.e. aii = 0 for all i). The most obvious example is the real skew-symmetric matrix. Other examples are the adjacency matrix of a finite simple graph, and a distance matrix or Euclidean distance matrix. In other words, any square matrix that takes the form is a hollow matrix, where the symbol \ast denotes an arbitrary entry. For example, is a hollow matrix.

Properties

e into the complement of the span of e . That is, where