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.

翻译：收敛性是迭代算法中的关键问题。阻尼常被用于确保迭代算法的收敛性。传统阻尼方法多为标量方式，且依赖于启发式或经验性策略。近期，有研究针对记忆型消息传递（迭代）算法提出了一种解析优化的向量阻尼方法，由此产生了一类特殊的协方差矩阵——L-带状矩阵。本文揭示了这类矩阵因其L-带状结构而具有的广泛代数性质。具体而言，我们推导了LDL分解、Cholesky分解、列替换后的行列式、子式及余子式的紧凑解析表达式。此外，给出了L-带状矩阵正定性的充要条件、特征多项式的递推求解方法以及其他若干性质。同时，我们提供了行列式和逆矩阵的新推导方法。