hollow matrix

{{short description|Several types of mathematical matrix containing zeroes}}

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.{{cite book | author=Pierre Massé | title=Optimal Investment Decisions: Rules for Action and Criteria for Choice | publisher=Prentice-Hall | year=1962 | page=142 }}

=Block of zeroes=

A hollow matrix may be a square {{math|n × n}} matrix with an {{math|r × s}} block of zeroes where {{math|r + s > n}}.{{cite book | author=Paul Cohn | authorlink=Paul Cohn | title=Free Ideal Rings and Localization in General Rings | url=https://archive.org/details/freeidealringslo00cohn | url-access=limited | publisher=Cambridge University Press | year=2006 | isbn=0-521-85337-0 | page=[https://archive.org/details/freeidealringslo00cohn/page/n453 430] }}

=Diagonal entries all zero=

A hollow matrix may be a square matrix whose diagonal elements are all equal to zero.{{cite book | author=James E. Gentle | title=Matrix Algebra: Theory, Computations, and Applications in Statistics | publisher=Springer-Verlag | year=2007 | isbn=978-0-387-70872-0 | page=42 |url=https://books.google.com/books?id=PDjIV0iWa2cC&q=%22hollow+matrix%22}} That is, an {{math|n × n}} matrix {{math|1=A = (a_ij)}} is hollow if {{math|1=a_ij = 0}} whenever {{math|1=i = j}} (i.e. {{math|1=a_ii = 0}} for all {{mvar|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

$\begin{pmatrix}$

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 $\ast$ denotes an arbitrary entry.

For example,

$\begin{pmatrix}$

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 {{mvar|A}} represents a linear map $L:V \to V$ with respect to a fixed basis, then it maps each basis vector {{math|e}} into the complement of the span of {{math|e}}. That is, $L(\langle e \rangle) \cap \langle e \rangle = \langle 0 \rangle$ where $\langle e \rangle = \{ \lambda e : \lambda \in F\}.$
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

Category:Matrices (mathematics)