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.

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 $n × n$ matrix with an $r × s$ block of zeroes where $r + s > n$.

Diagonal entries all zero
A hollow matrix may be a square matrix whose diagonal elements are all equal to zero. That is, an $n × n$ matrix $A = (a_{ij})$ is hollow if $a_{ij} = 0$ whenever $i = j$ (i.e. $a_{ii} = 0$ 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 $$\begin{pmatrix} 0  & \ast & & \ast & \ast \\ \ast & 0   & & \ast & \ast \\ & & \ddots \\ \ast & \ast & & 0 & \ast \\ \ast & \ast & & \ast & 0 \end{pmatrix}$$ is a hollow matrix, where the symbol $$\ast$$ denotes an arbitrary entry.

For example, $$\begin{pmatrix} 0 & 2 & 6 & \frac{1}{3} & 4 \\ 2 & 0 & 4 & 8 & 0 \\ 9 & 4 &  0 & 2 & 933 \\ 1 & 4 &  4 & 0 & 6 \\ 7 & 9 & 23 & 8 & 0 \end{pmatrix}$$ is a hollow matrix.

Properties

 * The trace of a hollow matrix is zero.
 * If $A$ represents a linear map $$L:V \to V$$with respect to a fixed basis, then it maps each basis vector $e$ into the complement of the span of $e$. That is, $$L(\langle e \rangle) \cap \langle e \rangle = \langle 0 \rangle$$ where $$\langle e \rangle = \{ \lambda e : \lambda \in F\}.$$
 * 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.