Toeplitz matrices and convergence

We investigate ${\displaystyle \left||{\chi }_A,2|\right|}$, the minimum cardinality of a subset of ${\displaystyle 2^{\omega }}$ that cannot be made convergent by multiplication with a single matrix taken from ${\displaystyle A}$, for different sets ${\displaystyle A}$ of Toeplitz matrices, and show that for some sets ${\displaystyle A}$ it coincides with the splitting number. We show that there is no Galois-Tukey connection from the chaos relation on the diagonal matrices to the chaos relation on the Toeplitz matrices with the identity on ${\displaystyle 2^{\omega }}$ as first component. With Suslin c.c.c. forcing we show that ${\displaystyle \left||{\chi }_M,2|\right|}$ < ${\displaystyle gb\bullet gs}$ is consistent relative to ZFC.