Hosta

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 $n \times n$ matrix with an $r \times s$ 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 $n \times n$ 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

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

^ Pierre Massé (1962). Optimal Investment Decisions: Rules for Action and Criteria for Choice. Prentice-Hall. p. 142.
^ Paul Cohn (2006). Free Ideal Rings and Localization in General Rings. Cambridge University Press. p. 430. ISBN 0-521-85337-0.
^ James E. Gentle (2007). Matrix Algebra: Theory, Computations, and Applications in Statistics. Springer-Verlag. p. 42. ISBN 978-0-387-70872-0.

[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.

[1]

[2]

[3]

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