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Hollow matrix
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<h2 id="definitions">Definitions</h2> <h3>Sparse</h3> <p>A <i>hollow matrix</i> may be one with "few" non-zero entries: that is, a <a href="/facts/Sparse_matrix/FFtN0XjR">sparse matrix</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> <h3>Block of zeroes</h3> <p>A <i>hollow matrix</i> may be a square <i>n</i> × <i>n</i> matrix with an <i>r</i> × <i>s</i> block of zeroes where <i>r</i> + <i>s</i> > <i>n</i>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> <h3>Diagonal entries all zero</h3> <p>A <i>hollow matrix</i> may be a <a href="/facts/Square_matrix/5TP9mNNn">square matrix</a> whose <a href="/facts/Diagonal/39z9sIRb">diagonal</a> elements are all equal to zero.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> That is, an <i>n</i> × <i>n</i> matrix <i>A</i> = (<i>aij</i>) is hollow if <i>aij</i> = 0 whenever <i>i</i> = <i>j</i> (i.e. <i>aii</i> = 0 for all i). The most obvious example is the <a href="/facts/Real_numbers/R02gw5Pb">real</a> <a href="/facts/Skew-symmetric_matrix/nw5CQeLN">skew-symmetric</a> matrix. Other examples are the <a href="/facts/Adjacency_matrix/jYXNhYGs">adjacency matrix</a> of a finite <a href="/facts/Simple_graph/kw3eIBUe">simple graph</a>, and a <a href="/facts/Distance_matrix/2ptk07Y7">distance matrix</a> or <a href="/facts/Euclidean_distance_matrix/cv1M8Hlx">Euclidean distance matrix</a>. </p><p>In other words, any square matrix that takes the form ( 0 ∗ ∗ ∗ ∗ 0 ∗ ∗ ⋱ ∗ ∗ 0 ∗ ∗ ∗ ∗ 0 ) {\displaystyle {\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 ∗ {\displaystyle \ast } denotes an arbitrary entry. </p><p>For example, ( 0 2 6 1 3 4 2 0 4 8 0 9 4 0 2 933 1 4 4 0 6 7 9 23 8 0 ) {\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}}} is a hollow matrix. </p> <h2 id="properties">Properties</h2> <ul><li>The <a href="/facts/Trace_(linear_algebra)/dchFFvHt">trace</a> of a hollow matrix is zero.</li> <li>If A represents a <a href="/facts/Linear_map/Q1PBKNVV">linear map</a> L : V → V {\displaystyle L:V\to V} with respect to a fixed <a href="/facts/Basis_(linear_algebra)/89IPoN6c">basis</a>, then it maps each basis vector e into the <a href="/facts/Complement_(set_theory)/yWUePIYY">complement</a> of the <a href="/facts/Linear_span/vPjclEtb">span</a> of e. That is, L ( ⟨ e ⟩ ) ∩ ⟨ e ⟩ = ⟨ 0 ⟩ {\displaystyle L(\langle e\rangle )\cap \langle e\rangle =\langle 0\rangle } where ⟨ e ⟩ = { λ e : λ ∈ F } . {\displaystyle \langle e\rangle =\{\lambda e:\lambda \in F\}.} </li> <li>The <a href="/facts/Gershgorin_circle_theorem/tXde6pGo">Gershgorin circle theorem</a> shows that the moduli of the <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalues</a> of a hollow matrix are less or equal to the sum of the moduli of the non-diagonal row entries.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Pierre Massé (1962). Optimal Investment Decisions: Rules for Action and Criteria for Choice. Prentice-Hall. p. 142. <a href="/wiki/Prentice-Hall" target="_blank">/wiki/Prentice-Hall</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Paul Cohn (2006). Free Ideal Rings and Localization in General Rings. Cambridge University Press. p. 430. ISBN 0-521-85337-0. <a href="0-521-85337-0" target="_blank">0-521-85337-0</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>James E. Gentle (2007). Matrix Algebra: Theory, Computations, and Applications in Statistics. Springer-Verlag. p. 42. ISBN 978-0-387-70872-0. <a href="978-0-387-70872-0" target="_blank">978-0-387-70872-0</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> </ol>