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

is a hollow matrix, where the symbol denotes an arbitrary entry.

For example,

is a hollow matrix.

Properties

• The trace of a hollow matrix is zero.
• If represents a linear map with respect to a fixed basis, then it maps each basis vector into the complement of the span of . That is, where
• 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