  We provide several new characterizations of $A_{p,\infty}$-matrix weights,
originally introduced by A. Volberg as matrix-valued substitutes of the
classical $A_\infty$ weights. In analogy with the notion of $A_p$-dimension of
matrix weights introduced in our previous work, we introduce the concepts of
the lower and the upper dimensions of $A_{p,\infty}$-matrix weights, which
enable us to obtain sharp estimates related to their reducing operators. In a
follow-up work, these results will play a key role in the study of function
spaces with $A_{p,\infty}$-matrix weights, which extends earlier results in the
more restricted class of $A_p$-matrix weights.
