Hollow matrix

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 = (a_ij) is hollow if a_ij = 0 whenever i = j (i.e. a_ii = 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 $${\displaystyle {\begin{pmatrix}0&\ast &&\ast &\ast \\\ast &0&&\ast &\ast \\&&\ddots \\\ast &\ast &&0&\ast \\\ast &\ast &&\ast &0\end{pmatrix}}}$$ is a hollow matrix, where the symbol ${\displaystyle \ast }$ denotes an arbitrary entry.

For example, $${\displaystyle {\begin{pmatrix}0&2&6&{\frac {1}{3}}&4\\2&0&4&8&0\\9&4&0&2&933\\1&4&4&0&6\\7&9&23&8&0\end{pmatrix}}}$$ is a hollow matrix.

Properties

• The trace of a hollow matrix is zero.
• If A represents a linear map ${\displaystyle L:V\to V}$with respect to a fixed basis, then it maps each basis vector e into the complement of the span of e. That is, ${\displaystyle L(\langle e\rangle )\cap \langle e\rangle =\langle 0\rangle }$ where ${\displaystyle \langle e\rangle =\{\lambda e:\lambda \in F\}.}$
• The Gershgorin circle theorem shows that the moduli of the eigenvalues of a hollow matrix are less or equal to the sum of the moduli of the non-diagonal row entries.

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.