$ \boldsymbol{M} $-matrices are an important class of special matrices for numerical modeling in biology, physics, and other disciplines. Estimation of the minimum eigenvalue of the Schur product of $ \boldsymbol{M} $-matrices is a significant research topic in the field of matrix analysis. Based on the Gerschgorin disk theorem and the Brauer oval theorem, this paper established new lower bound estimates for the minimum eigenvalue of the Schur product of an $ \boldsymbol{M} $-matrix and an inverse $ \boldsymbol{M} $-matrix. The proposed bounds only rely on the entries of the matrices, with compact expressions and high computational efficiency. A series of numerical examples illustrate that the bounds derived in this paper are tighter than several estimation formulas from existing literature, serving as a valuable supplement to the theory of spectral bound estimation for Schur products of $ \boldsymbol{M} $-matrices.
Citation: Xiaofeng Zhang, Fubin Chen. New bounds for the eigenvalue of the Schur product of an $ {{M}} $-matrix and inverse $ {{M}} $-matrix[J]. AIMS Mathematics, 2026, 11(9): 31396-31411. doi: 10.3934/math.20261239
$ \boldsymbol{M} $-matrices are an important class of special matrices for numerical modeling in biology, physics, and other disciplines. Estimation of the minimum eigenvalue of the Schur product of $ \boldsymbol{M} $-matrices is a significant research topic in the field of matrix analysis. Based on the Gerschgorin disk theorem and the Brauer oval theorem, this paper established new lower bound estimates for the minimum eigenvalue of the Schur product of an $ \boldsymbol{M} $-matrix and an inverse $ \boldsymbol{M} $-matrix. The proposed bounds only rely on the entries of the matrices, with compact expressions and high computational efficiency. A series of numerical examples illustrate that the bounds derived in this paper are tighter than several estimation formulas from existing literature, serving as a valuable supplement to the theory of spectral bound estimation for Schur products of $ \boldsymbol{M} $-matrices.
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Xiaofeng Zhang, Fubin Chen. New bounds for the eigenvalue of the Schur product of an $ {{M}} $-matrix and inverse $ {{M}} $-matrix[J]. AIMS Mathematics, 2026, 11(9): 31396-31411. doi: 10.3934/math.20261239
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