$M^{*}$-matrices: Tridiagonal structure and signature similarity to $M$-matrices

Source abstract

This paper introduces and studies a new class of matrices, called $M^{*}$-matrices. These matrices have a prescribed tridiagonal sign structure together with the positivity of all principal minors. Specifically, $M^{*}$-matrices are matrices with positive diagonal entries, positive first sub- and super-diagonal entries, non-positive entries outside the tridiagonal band, and positive principal minors. The specified sign structure and characteristics of $M^*$-matrices extend both classical tridiagonal $M$-matrices and totally positive tridiagonal matrices. We identify the subclass, denoted by $M_T^{*}$, consisting of all tridiagonal $M^{*}$-matrices, such that each matrix in this class is always diagonally signature-similar to a tridiagonal $M$-matrix. The classical $M$-matrix theory can be transferred directly to this subclass. In particular, $M_T^{*}$-matrices admit $LU$ factorizations without pivoting with positive pivots, have eigenvalues with positive real parts, exhibit inverse-sign regularity, and possess invariance of induced norm condition numbers. The symmetric $M_T^{*}$-matrices coincide with Jacobi matrices and are therefore positive definite, admitting Cholesky factorizations and strict eigenvalue interlacing. We also investigate scaling invariance and numerical conditioning. Applications of this structured tridiagonal system arise in compartmental and population models with alternating nearest-neighbor transfer. This paper illustrates the natural occurrence of $M^{*}$-matrices in applied and computational settings. Received: 07 March 2026 | Accepted: 30 April 2026