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Hollow matrix

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

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

$begin\left\{array\right\}\left\{rlll\right\}$
A_{ntimes n} & = & (a_{ij}); a_{ij} & = & 0 & mbox{if} quad i=j,quad 1le i,j le n., end{array}

In other words, any square matrix which takes the form $left\left(begin\left\{array\right\}\left\{ccccc\right\} 0 & 0 & & ddots & & & 0 & & & & 0end\left\{array\right\}right\right)$  is a hollow matrix.

For example: $left\left(begin\left\{array\right\}\left\{ccccc\right\} 0 & 2 & 6 & frac\left\{1\right\}\left\{3\right\} & 42 & 0 & 4 & 8 & 0 9 & 4 & 0 & 2 & 933 1 & 4 & 4 & 0 & 6 7 & 9 & 23 & 8 & 0end\left\{array\right\}right\right)$  is an example of a hollow matrix.

Properties

• The trace of A is trivially zero.

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

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