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H-matrix (iterative method)
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<p>In mathematics, an <i>H</i>-matrix is a <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> whose <a href="/facts/Comparison_matrix/LHvezL1k">comparison matrix</a> is an <a href="/facts/M-matrix/qsoroftJ">M-matrix</a>. It is useful in <a href="/facts/Iterative_method/uKMAJhEh">iterative methods</a>. </p><p>Definition: Let <i>A</i> = (<i>aij</i>) be a <i>n</i> × <i>n</i> complex matrix. Then <a href="/facts/Comparison_matrix/LHvezL1k">comparison matrix</a> <i>M</i>(<i>A</i>) of complex matrix <i>A</i> is defined as <i>M</i>(<i>A</i>) = <i>α</i><i>ij</i> where <i>α</i><i>ij</i> = −|<i>A</i><i>ij</i>| for all <i>i</i> ≠ <i>j</i>, 1 ≤ <i>i,j</i> ≤ <i>n</i> and <i>αij</i> = |<i>Aij</i>| for all <i>i</i> = <i>j</i>, 1 ≤ <i>i,j</i> ≤ <i>n</i>. If <i>M</i>(<i>A</i>) is a <a href="/facts/M-matrix/qsoroftJ">M-matrix</a>, <i>A</i> is a <i>H</i>-matrix. </p><p><a href="/facts/Invertible_matrix/fPqXk3V8">Invertible</a> H-matrix guarantees convergence of <a href="/facts/Gauss%25E2%2580%2593Seidel_method/eK94QeFt">Gauss–Seidel iterative methods</a>. </p>