论文标题
L带矩阵的代数
Algebra of L-banded Matrices
论文作者
论文摘要
收敛是迭代算法中的关键问题。通常使用阻尼来确保迭代算法的收敛性。常规的阻尼方式是标量的，是启发式或经验性的。最近，提出了一种分析优化的向量阻尼，用于记忆消息通话（迭代）算法。结果，它产生了一种称为L带矩阵的特殊协方差矩阵。在本文中，我们表明这些矩阵具有由L带结构产生的宽大代数特性。特别是，得出了LDL分解的紧凑分析表达式，Cholesky分解，柱取代后的决定因素，未成年人和辅因子。此外，要确定L带矩阵的必要条件和足够的条件，以获得特征多项式的复发性，并给出了其他一些特性。此外，我们给出了决定因素和逆的新推导。 （至关重要的是要强调，有些作品已经独立研究了具有这种特殊结构的矩阵，称为l-matrices。具体来说，l带矩阵被视为具有真实和有限条目的L型矩阵）。
Convergence is a crucial issue in iterative algorithms. Damping is commonly employed to ensure the convergence of iterative algorithms. The conventional ways of damping are scalar-wise, and either heuristic or empirical. Recently, an analytically optimized vector damping was proposed for memory message-passing (iterative) algorithms. As a result, it yields a special class of covariance matrices called L-banded matrices. In this paper, we show these matrices have broad algebraic properties arising from their L-banded structure. In particular, compact analytic expressions for the LDL decomposition, the Cholesky decomposition, the determinant after a column substitution, minors, and cofactors are derived. Furthermore, necessary and sufficient conditions for an L-banded matrix to be definite, a recurrence to obtain the characteristic polynomial, and some other properties are given. In addition, we give new derivations of the determinant and the inverse. (It's crucial to emphasize that some works have independently studied matrices with this special structure, named as L-matrices. Specifically, L-banded matrices are regarded as L-matrices with real and finite entries.)
