Hollow Matrix

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.

Block of zeroes

A ''hollow matrix'' may be a square matrix with an block of zeroes where .

Diagonal entries all zero

A ''hollow matrix'' may be a square matrix whose diagonal elements are all equal to zero. 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.
*If represents a linear map $L:V \to V$with respect to a fixed basis, then it maps each basis vector into the complement of the span of . That is, $L(\langle e \rangle) \cap \langle e \rangle = \langle 0 \rangle$ where $\langle e \rangle = \.$
*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

Matrices (mathematics)

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