Hollow matrix

Last updated November 30, 2023

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.

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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.

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[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.

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v t e Matrix classes
Explicitly constrained entries	Alternant Anti-diagonal Anti-Hermitian Anti-symmetric Arrowhead Band Bidiagonal Bisymmetric Block-diagonal Block Block tridiagonal Boolean Cauchy Centrosymmetric Conference Complex Hadamard Copositive Diagonally dominant Diagonal Discrete Fourier Transform Elementary Equivalent Frobenius Generalized permutation Hadamard Hankel Hermitian Hessenberg Hollow Integer Logical Matrix unit Metzler Moore Nonnegative Pentadiagonal Permutation Persymmetric Polynomial Quaternionic Signature Skew-Hermitian Skew-symmetric Skyline Sparse Sylvester Symmetric Toeplitz Triangular Tridiagonal Vandermonde Walsh Z
Constant	Exchange Hilbert Identity Lehmer Of ones Pascal Pauli Redheffer Shift Zero
Conditions on eigenvalues or eigenvectors	Companion Convergent Defective Definite Diagonalizable Hurwitz Positive-definite Stieltjes
Satisfying conditions on products or inverses	Congruent Idempotent or Projection Invertible Involutory Nilpotent Normal Orthogonal Unimodular Unipotent Unitary Totally unimodular Weighing
With specific applications	Adjugate Alternating sign Augmented Bézout Carleman Cartan Circulant Cofactor Commutation Confusion Coxeter Distance Duplication and elimination Euclidean distance Fundamental (linear differential equation) Generator Gram Hessian Householder Jacobian Moment Payoff Pick Random Rotation Seifert Shear Similarity Symplectic Totally positive Transformation
Used in statistics	Centering Correlation Covariance Design Doubly stochastic Fisher information Hat Precision Stochastic Transition
Used in graph theory	Adjacency Biadjacency Degree Edmonds Incidence Laplacian Seidel adjacency Tutte
Used in science and engineering	Cabibbo–Kobayashi–Maskawa Density Fundamental (computer vision) Fuzzy associative Gamma Gell-Mann Hamiltonian Irregular Overlap S State transition Substitution Z (chemistry)
Related terms	Jordan normal form Linear independence Matrix exponential Matrix representation of conic sections Perfect matrix Pseudoinverse Row echelon form Wronskian
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