Hollow matrix

In mathematics, a hollow matrix may refer to one of several related classes of matrix.

Sparse

A hollow matrix may be one with "few" non-zero entries: that is, a sparse matrix.^[1]

Diagonal entries all zero

A hollow matrix may be a square matrix whose diagonal elements are all equal to zero .^[2] The most obvious example is the real skew-symmetric matrix. Other examples are the adjacency matrix of a finite simple graph; a distance matrix or Euclidean distance matrix.

If A is an n×n hollow matrix, then the elements of A are given by

${\begin{array}{rlll}A_{{n\times n}}&=&(a_{{ij}});\\a_{{ij}}&=&0&{\mbox{if}}\quad i=j,\quad 1\leq i,j\leq n.\,\end{array}}$

In other words, any square matrix which takes the form $\left({\begin{array}{ccccc}0\\&0\\&&\ddots \\&&&0\\&&&&0\end{array}}\right)$  is a hollow matrix.

For example: $\left({\begin{array}{ccccc}0&2&6&{\frac {1}{3}}&4\\2&0&4&8&0\\9&4&0&2&933\\1&4&4&0&6\\7&9&23&8&0\end{array}}\right)$  is an example of a hollow matrix.

Properties

• The trace of A is trivially zero.
• The linear map represented by A (with respect to a fixed basis) maps each basis vector e onto the image of the complement of <e>.
• Gershgorin's Circle Theorem shows that the moduli of the eigenvalues of A are less or equal to the sum of the moduli of the non-diagonal row entries.

Block of zeroes

A hollow matrix may be a square n×n matrix with an r×s block of zeroes where r+s>n.^[3]

References

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