This paper proposes a block-tridiagonal state-space representation to establish alternative algebraic sufficient conditions for the absolute stability of multivariable non-linear systems. The systems considered are described by a transfer function matrix (given by a left matrix-fraction description) with multiple non-linear feedback elements satisfying the positive infinite sector condition. New matrix algorithms are presented in order to construct the block-tridiagonal representation of the system. The relationship of these absolute stability conditions to passivity properties is discussed.