  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.)
