Several types of mathematical matrix containing zeroes
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.[1]
Block of zeroes
A hollow matrix may be a square n × n matrix with an r × s block of zeroes where r + s > n.[2]
Diagonal entries all zero
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
![{\displaystyle {\begin{pmatrix}0&\ast &&\ast &\ast \\\ast &0&&\ast &\ast \\&&\ddots \\\ast &\ast &&0&\ast \\\ast &\ast &&\ast &0\end{pmatrix))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dacd4019832e1a85070ccd55965f887efb8f78aa)
is a hollow matrix, where the symbol
denotes an arbitrary entry.
For example,
![{\displaystyle {\begin{pmatrix}0&2&6&{\frac {1}{3))&4\\2&0&4&8&0\\9&4&0&2&933\\1&4&4&0&6\\7&9&23&8&0\end{pmatrix))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1d057c9921b2b17366323b192e6a532258a8ae6)
is a hollow matrix.
Properties
- The trace of a hollow matrix is zero.
- If A represents a linear map
with respect to a fixed basis, then it maps each basis vector e into the complement of the span of e. That is,
where ![{\displaystyle \langle e\rangle =\{\lambda e:\lambda \in F\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0f16d9a452cd6bc0a671d6885da6dbfcde7e20b)
- 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.