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.

Definitions

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 Template:Math matrix with an Template:Math block of zeroes where Template:Math.

Diagonal entries all zero

A hollow matrix may be a square matrix whose diagonal elements are all equal to zero. That is, an Template:Math matrix Template:Math is hollow if Template:Math whenever Template:Math (i.e. Template:Math for all Template:Mvar). 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 $$\left(\begin{array}{ccccc}0& \ast & & \ast & \ast \\ \ast & 0& & \ast & \ast \\ & & \ddots \\ \ast & \ast & & 0& \ast \\ \ast & \ast & & \ast & 0\end{array}\right)$$ is a hollow matrix, where the symbol $\ast$ denotes an arbitrary entry.

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

Properties

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

References

Template:Matrix-stub