The spectral radius of a matrix is widely used in numerical analysis, graph theory, stability theory, and other related fields, and is a rather active topic in matrix theory research. In this paper, we establish a smoothing algorithm to calculate the spectral radius of a non-detective nonnegative irreducible matrix by constructing a special matrix. The effectiveness of the algorithm is demonstrated by a numerical arithmetic example.
The Hadamard product of matrices is widely used in image processing and blind source signal separation and is a hot topic in matrix analysis theory. For nonnegative matrices with nonzero principal diagonal elements, a new upper bound of the spectral radius of the Hadamard product is established by using the eigenvalue inclusion field theorem and a special similar diagonalization operation. The new estimation can be expressed only by the elements of two nonnegative matrices, which is operable and easy to calculate. Finally, an example is given to verify that the estimation formula in this paper is more accurate than that in related research.
Based on the classic eigenvalue inclusion theorem, we establish a new lower bound expression on the minimum eigenvalue for the Fan product of two M-matrices. The new bound is only related to the elements of two M-matrices and is more operative in practical applications. In addition, a numerical example is considered to demonstrate the feasibility and validity of the result obtained.
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