Hollow matrix explained

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 matrix with an block of zeroes where .^[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 matrix is hollow if whenever (i.e. for all). 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 $\begin 0 & \ast & & \ast & \ast \\\ast & 0 & & \ast & \ast \\ & & \ddots \\\ast & \ast & & 0 & \ast \\\ast & \ast & & \ast & 0\end$ is a hollow matrix, where the symbol

\ast

denotes an arbitrary entry.

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

Properties

The trace of a hollow matrix is zero.

L:V\toV

with respect to a fixed basis, then it maps each basis vector into the complement of the span of . That is,

L(\langlee\rangle)\cap\langlee\rangle=\langle0\rangle

where

\langlee\rangle=\{λe:λ\inF\}.

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.

Notes and References

Book: Pierre Massé . Optimal Investment Decisions: Rules for Action and Criteria for Choice . . 1962 . 142 .
Book: Paul Cohn
. Paul Cohn . Paul Cohn . Free Ideal Rings and Localization in General Rings . limited . . 2006 . 0-521-85337-0 . 430 .
Book: James E. Gentle . Matrix Algebra: Theory, Computations, and Applications in Statistics . . 2007 . 978-0-387-70872-0 . 42 .