Hollow matrix

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

Definitions

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

${\displaystyle {\begin{aligned}A_{n\times n}&=(a_{ij}),\\a_{ij}&=0\quad {\text{if}}\ i=j,\ 1\leq i,j\leq n.\end{aligned}}}$

In other words, any square matrix that takes the form

${\displaystyle {\begin{pmatrix}0\\&0\\&&\ddots \\&&&0\\&&&&0\end{pmatrix}}}$

is a hollow matrix.

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 is zero.
• If A represents a linear operator ${\displaystyle L:V\to V}$with respect to a fixed basis, then it maps each basis vector e into the complement of the span of e, i.e. ${\displaystyle L(e)\cap \langle e\rangle =\emptyset }$where ${\displaystyle \langle e\rangle =\{\lambda e:\lambda \in F\}}$
• Gershgorin 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.