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.
A hollow matrix may be one with "few" non-zero entries: that is, a sparse matrix.
[1] A hollow matrix may be a square n × n matrix with an r × s block of zeroes where r + s > n.[2] 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
is a hollow matrix, where the symbol
denotes an arbitrary entry.
is a hollow matrix.
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