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.)

翻译：收敛性是迭代算法中的关键问题。阻尼技术常被用于确保迭代算法的收敛性。传统的阻尼方式基于标量运算，且多为启发式或经验性方法。近期，针对记忆型消息传递（迭代）算法提出了一种经解析优化的向量阻尼方法，由此导出了一类特殊的协方差矩阵——L-带状矩阵。本文证明，这类矩阵因其L-带状结构而具有广泛的代数性质。具体而言，我们推导了LDL分解、Cholesky分解、列替换后的行列式、子式及余子式的紧致解析表达式。此外，给出了L-带状矩阵正定性的充要条件、特征多项式递推公式及其他性质，同时提出了行列式与逆矩阵的新推导方法。（需特别说明的是，已有独立研究关注此类特殊结构矩阵，并称之为L-矩阵。具体而言，L-带状矩阵可视为元素为实数且有界的L-矩阵。）