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.[1]

Block of zeroes

A hollow matrix may be a square n × n matrix with an r × s block of zeroes where r + s > n.[2]

Diagonal entries all zero

A hollow matrix may be a square matrix whose diagonal elements are all equal to zero.[3] 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 denotes an arbitrary entry.

For example,

is a hollow matrix.

Properties

References

  1. ^ Pierre Massé (1962). Optimal Investment Decisions: Rules for Action and Criteria for Choice. Prentice-Hall. p. 142.
  2. ^ Paul Cohn (2006). Free Ideal Rings and Localization in General Rings. Cambridge University Press. p. 430. ISBN 0-521-85337-0.
  3. ^ James E. Gentle (2007). Matrix Algebra: Theory, Computations, and Applications in Statistics. Springer-Verlag. p. 42. ISBN 978-0-387-70872-0.