Inmathematics, ahollow matrix may refer to one of several related classes ofmatrix: a sparse matrix; a matrix with a large block of zeroes; or a matrix with diagonal entries all zero.

Ahollow matrix may be one with "few" non-zero entries: that is, asparse matrix.^[1]

Ahollow matrix may be a squaren ×n matrix with anr ×s block of zeroes wherer +s >n.^[2]

Ahollow matrix may be asquare matrix whosediagonal elements are all equal to zero.^[3] That is, ann ×n matrixA = (a_ij) is hollow ifa_ij = 0 wheneveri =j (i.e.a_ii = 0 for alli). The most obvious example is therealskew-symmetric matrix. Other examples are theadjacency matrix of a finitesimple graph, and adistance matrix orEuclidean distance matrix.

In other words, any square matrix that takes the formis a hollow matrix, where the symbol${\displaystyle \ast }$ denotes an arbitrary entry.

For example,is a hollow matrix.

• Thetrace of a hollow matrix is zero.
• IfA represents alinear map${\displaystyle L:V\to V}$with respect to a fixedbasis, then it maps each basis vectore into thecomplement of thespan ofe. That is,${\displaystyle L(⟨e⟩)\cap ⟨e⟩=⟨0⟩}$ where${\displaystyle ⟨e⟩=\{\lambda e:\lambda \in F\}.}$
• TheGershgorin circle theorem shows that the moduli of theeigenvalues of a hollow matrix are less or equal to the sum of the moduli of the non-diagonal row entries.

• Matrices (mathematics)

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