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Hollow matrix

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In mathematics, a hollow matrix may refer to one of several related classes of matrix.

Contents

Sparse

A hollow matrix may be one with "few" non-zero entries: that is, a sparse matrix.

Diagonal entries all zero

A hollow matrix may be a square matrix whose diagonal elements are all equal to zero. The most obvious example is the real skew-symmetric matrix. Other examples are the adjacency matrix of a finite simple graph; a distance matrix or Euclidean distance matrix.

If A is an n×n hollow matrix, then the elements of A are given by

A n × n = ( a i j ) , a i j = 0 if   i = j ,   1 ≤ i , j ≤ n .

In other words, any square matrix that takes the form

( 0 0 ⋱ 0 0 )

is a hollow matrix.

For example:

( 0 2 6 1 3 4 2 0 4 8 0 9 4 0 2 933 1 4 4 0 6 7 9 23 8 0 )

is a hollow matrix.

Properties

  • The trace of A is trivially zero.
  • The linear map represented by A (with respect to a fixed basis) maps each basis vector e onto the image of the complement of span(e).
  • Gershgorin circle theorem shows that the moduli of the eigenvalues of A are less or equal to the sum of the moduli of the non-diagonal row entries.

    • Block of zeroes

    A hollow matrix may be a square n×n matrix with an r×s block of zeroes where r+s>n.

    References

    Hollow matrix Wikipedia
    (Text) CC BY-SA


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