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.[1]

Block of zeroes

A hollow matrix may be a square n × n matrix with an r × 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 × n matrix A = (aij) is hollow if aij = 0 whenever i = j (i.e. aii = 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 $\displaystyle{ \begin{pmatrix} 0 & \ast & & \ast & \ast \\ \ast & 0 & & \ast & \ast \\ & & \ddots \\ \ast & \ast & & 0 & \ast \\ \ast & \ast & & \ast & 0 \end{pmatrix} }$ is a hollow matrix, where the symbol $\displaystyle{ \ast }$ denotes an arbitrary entry.

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

1. Pierre Massé (1962). Optimal Investment Decisions: Rules for Action and Criteria for Choice. Prentice-Hall. p. 142.