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<p>The power method is a basic method for computing the dominant eigenpair of a matrix. In this paper, we propose a structure-preserving power-like method for computing the dominant conjugate pair of purely imaginary eigenvalues and the corresponding eigenvectors of a large skew-symmetric matrix <i>S</i>, which works on <i>S</i> and its transpose <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(S^{T}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>S</mi> <mi>T</mi> </msup> </math></EquationSource> </InlineEquation> alternately and is performed in <i>real</i> arithmetic. We derive quantitative convergence results on the approximate dominant eigenvalue and eigenvector obtained by the ordinary power method when the underlying matrix is normal, based on which we establish the rigorous and quantitative convergence results of the proposed power-like method, and prove that the approximations to the dominant eigenvalues converge twice as fast as those to the associated eigenvectors. Moreover, we develop a deflation technique to compute several complex conjugate dominant eigenpairs of <i>S</i>. Numerical experiments show the effectiveness and efficiency of the new method.</p>

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A Power-like Method for Computing the Dominant Eigenpairs of Large Scale Real Skew-Symmetric Matrices

  • 期刊论文

  • 发表日期：

    2025年11月3日

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  • Qingqing Zheng

摘要

The power method is a basic method for computing the dominant eigenpair of a matrix. In this paper, we propose a structure-preserving power-like method for computing the dominant conjugate pair of purely imaginary eigenvalues and the corresponding eigenvectors of a large skew-symmetric matrix S, which works on S and its transpose \(S^{T}\) S T alternately and is performed in real arithmetic. We derive quantitative convergence results on the approximate dominant eigenvalue and eigenvector obtained by the ordinary power method when the underlying matrix is normal, based on which we establish the rigorous and quantitative convergence results of the proposed power-like method, and prove that the approximations to the dominant eigenvalues converge twice as fast as those to the associated eigenvectors. Moreover, we develop a deflation technique to compute several complex conjugate dominant eigenpairs of S. Numerical experiments show the effectiveness and efficiency of the new method.