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Several spectral radii formulas for infinite bounded non-negative matrices in max algebra are obtained. We also prove some Perron–Frobenius type results for such matrices. In particular, we obtain results on block triangular forms, which are similar to results on Frobenius normal form of $ n \times n $ n × n matrices. Some continuity results are also established. [ABSTRACT FROM AUTHOR]

Inspired by the idea and technique in a recent paper of Huy et al. (Linear Algebra Appl 656: 368–384, 2023), Ighachane and Bouchangour showed an important refinement of the results by Sababheh (Linear Algebra Appl 506: 588–602, 2016) involving both convex and log-convex functions. In this paper, we continue to establish further new inequalities for both convex and log-convex functions with some applications to scalar and matrix means as well as matrix norm and numerical radius. The significance of the obtained results is two folded: the results themselves and the way they extend many known results in the literature. [ABSTRACT FROM AUTHOR]

We prove new inequalities and equalities for the generalized and the joint spectral radius (and their essential versions) of Hadamard (Schur) geometric means of bounded sets of positive kernel operators on Banach function spaces. In the case of non-negative matrices that define operators on Banach sequences, we obtain additional results. Our results extend the results of several authors that appeared relatively recently. [ABSTRACT FROM AUTHOR]

The numerical radius of a matrix is a scalar quantity that has many applications in the study of matrix analysis. Due to the difficulty in computing the numerical radius, inequalities bounding it have received a considerable attention in the literature. In this article, we present many new inequalities for the numerical radius of accretive matrices. The importance of this study is the presence of a new approach that treats a specific class of matrices, namely the accretive ones. While some of these inequalities can be considered as refinements of other existing ones, others present new insight to some known results for positive matrices. [ABSTRACT FROM AUTHOR]

Reciprocal matrices and, in particular, transitive matrices, appear in several applied areas. Among other applications, they have an important role in decision theory in the context of the analytical hierarchical process, introduced by Saaty. In this paper, we study the possible ranks of a reciprocal matrix and give a procedure to construct a reciprocal matrix with the rank and the off-diagonal entries of an arbitrary row (column) prescribed. We apply some techniques from graph theory to the study of transitive matrices, namely to determine the maximum number of equal entries, and distinct from ± 1 , in a transitive matrix. We then focus on the n-by-n reciprocal matrix, denoted by C(n, x), with all entries above the main diagonal equal to x > 0. We show that there is a Toeplitz transitive matrix and a transitive matrix preserving the maximum possible number of entries of C(n, x), whose distances to C(n, x), measured in the Frobenius norm, are smaller than the one of the transitive matrix proposed by Saaty, constructed from the right Perron eigenvector of C(n, x). We illustrate our results with some numerical examples. [ABSTRACT FROM AUTHOR]

Let Ψ and Σ be bounded sets of positive kernel operators on a Banach function space L. We prove several refinements of the known inequalities for the generalized spectral radius ρ and the joint spectral radius , where denotes the Hadamard (Schur) geometric mean of the sets Ψ and Σ. Furthermore, we prove that analogous inequalities hold also for the generalized essential spectral radius and the joint essential spectral radius in the case when L and its Banach dual have order continuous norms. [ABSTRACT FROM AUTHOR]

The Thompson metric provides key geometric insights in the study of non-linear matrix equations and in many optimization problems. However, knowing that an approximate solution is within d T units, in the Thompson metric, of the actual solution provides little insight into how good the approximation is as a matrix or vector approximation. That is, bounding the Thompson metric between an approximate and accurate solution to a problem does not provide obvious bounds either for the spectral or the Frobenius norm, both Schatten norms, of the difference between the approximation and accurate solution. This paper reports such an upper bound, namely that ∥ X − Y ∥ p ≤ 2 1 p e d − 1 e d max ∥ X ∥ p , ∥ Y ∥ p where ⋅ p denotes the Schatten p-norm and d denotes the Thompson metric between X and Y. Furthermore, a more geometric proof leads to a slightly better bound in the case of the Frobenius norm, ∥ X − Y ∥ 2 ≤ e d − 1 e 2 d + 1 ∥ X ∥ 2 2 + ∥ Y ∥ 2 2 ≤ 2 1 2 e d − 1 e 2 d + 1 max ∥ X ∥ p , ∥ Y ∥ p . [ABSTRACT FROM AUTHOR]

Let A and B be nonnegative square matrices of the same order. Denote by ‖ ⋅ ‖ and ρ ( ⋅ ) the spectral norm and the spectral radius respectively. We prove the following inequalities: ‖ A ∘ B ‖ ≤ ‖ A ∘ A ‖ 1 2 ‖ B ∘ B ‖ 1 2 ; ‖ A ∘ B ‖ ≤ ρ 1 2 ( A T B ∘ B T A ) ≤ ρ 1 2 ( A T B ∘ A T B ) ≤ ρ ( A T B ) , where ∘ denotes the Hadamard product. This interpolates the inequality ‖ A ∘ B ‖ ≤ ρ ( A T B ) due to Huang. Some spectral norm inequalities for an arbitrarily finite number of nonnegative square matrices are also obtained, which refine some other results of Huang. [ABSTRACT FROM AUTHOR]

We present several results describing the interplay between the max algebraic joint spectral radius (JSR) for compact sets of matrices and suitably defined matrix norms. In particular, we extend a classical result for the conventional algebra, showing that the max algebraic JSR can be described in terms of induced norms of the matrices in the set. We also show that for a set generating an irreducible semigroup (in a cone-theoretic sense), a monotone Barabanov norm always exists. This fact is then used to show that the max algebraic JSR is locally Lipschitz continuous on the space of compact irreducible sets of matrices with respect to the Hausdorff distance. We then prove that the max algebraic JSR is locally Hoelder continuous on the space of compact sets of nonnegative matrices. Finally, we prove a strict monotonicity property for the max algebraic JSR that echoes a fact for the classical JSR. The single matrix characterisation of the max algebraic JSR plays a vital role in our proofs. [ABSTRACT FROM AUTHOR]

Let Ψ 1 , … , Ψ m be bounded sets of positive kernel operators on a Banach function space L . We prove that for the generalized spectral radius ρ and the joint spectral radius ρ ˆ the inequalities ρ ( Ψ 1 ( 1 m ) ∘ ⋯ ∘ Ψ m ( 1 m ) ) ≤ ρ ( Ψ 1 Ψ 2 ⋯ Ψ m ) 1 m , ρ ˆ ( Ψ 1 ( 1 m ) ∘ ⋯ ∘ Ψ m ( 1 m ) ) ≤ ρ ˆ ( Ψ 1 Ψ 2 ⋯ Ψ m ) 1 m hold, where Ψ 1 ( 1 m ) ∘ ⋯ ∘ Ψ m ( 1 m ) denotes the Hadamard (Schur) geometric mean of the sets Ψ 1 , … , Ψ m . [ABSTRACT FROM AUTHOR]

It is known that every complex square matrix with nonnegative determinant is the product of positive semi-definite matrices. There are characterizations of matrices that require two or five positive semi-definite matrices in the product. However, the characterizations of matrices that require three or four positive semi-definite matrices in the product are lacking. In this paper, we give a complete characterization of these two types of matrices. With these results, we give an algorithm to determine whether a square matrix can be expressed as the product of k positive semi-definite matrices but not fewer, for k = 1 , 2 , 3 , 4 , 5 . [ABSTRACT FROM AUTHOR]

It is known that the numerical radius of the Hadamard product A ∘ B of two n -by- n matrices A and B is related to those of A and B by (a) w ( A ∘ B ) ≤ 2 w ( A ) w ( B ) , (b) w ( A ∘ B ) ≤ w ( A ) w ( B ) if one of A and B is normal, and (c) w ( A ∘ B ) ≤ ( max i ⁡ a i i ) w ( B ) if A = [ a i j ] i , j = 1 n is positive semidefinite. In this paper, we give complete characterizations of A and B for which the equality is attained. The matrices involved can be considered as elaborate generalizations of the equality-attaining A = [ 0 0 a 0 ] and B = [ 0 0 b 0 ] for (a), A = [ a 1 0 0 a 2 ] ( | a 1 | ≥ | a 2 | ) and B = [ w ( B ) ⁎ ⁎ ⁎ ] for (b), and A = [ a 1 a 3 a 2 a 4 ] ≥ 0 ( a 1 ≥ a 4 ) and B = [ w ( B ) ⁎ ⁎ ⁎ ] for (c). [ABSTRACT FROM AUTHOR]

Several characterizations are given for a square matrix that can be written as the product of two positive (semidefinite) contractions. Based on one of these characterizations, and the theory of alternating projections, a Matlab program is written to check the condition and construct the two positive contractions whose product equal to the given matrix, if they exist. [ABSTRACT FROM AUTHOR]

We present an extension of the Perron–Frobenius theory to the numerical ranges of semi-monic Perron–Frobenius polynomials, namely matrix polynomials of the form Q ( λ ) = λ m I − ( λ l A l + ⋯ + A 0 ) = λ m I − A ( λ ) , where the coefficients are entrywise nonnegative matrices. Our approach relies on the function β ↦ numerical radius A ( β ) and the infinite graph G m ( A 0 , … , A l ) . Our main result describes the cyclic distribution of the elements of the numerical range of Q ( ⋅ ) on the circles with radius β satisfying β m = numerical radius A ( β ) . [ABSTRACT FROM AUTHOR]

Recently Blondel, Nesterov and Protasov proved [1,2] that the finiteness conjecture holds for the generalized and the lower spectral radii of the sets of non-negative matrices with independent row/column uncertainty. We show that this result can be obtained as a simple consequence of the so-called hourglass alternative used in [3] , by the author and his companions, to analyze the minimax relations between the spectral radii of matrix products. Axiomatization of the statements that constitute the hourglass alternative makes it possible to define a new class of sets of positive matrices having the finiteness property, which includes the sets of non-negative matrices with independent row uncertainty. This class of matrices, supplemented by the zero and identity matrices, forms a semiring with the Minkowski operations of addition and multiplication of matrix sets, which gives means to construct new sets of non-negative matrices possessing the finiteness property for the generalized and the lower spectral radii. [ABSTRACT FROM AUTHOR]

We give new proofs of several fundamental results of spectral theory in max algebra. This includes the description of the spectrum in max algebra of a given non-negative matrix via local spectral radii, the spectral theorem and the spectral mapping theorem in max algebra. The latter result is also generalized to the setting of power series in max algebra by applying certain continuity properties of the spectrum in max algebra. Our methods enable us to obtain some related results for the usual spectrum of complex matrices and distinguished spectrum for non-negative matrices. [ABSTRACT FROM AUTHOR]

For-by-and-by-complex matricesand, it is known that the inequalityholds, whereanddenote, respectively, the numerical radius and the operator norm of a matrix. In this paper, we consider when this becomes an equality. We show that (1) ifand, then one of the following two conditions holds: (i)has a unitary part, and (ii)is completely nonunitary and the numerical rangeofis a circular disc centered at the origin, (2) iffor some,, then, and, moreover, the equality holds if and only ifis unitarily similar to the direct sum of the-by-Jordan blockand a matrixwith, and (3) ifis a nonnegative matrix with its real part (permutationally) irreducible, then, if and only if eitherorandis permutationally similar to a block-shift matrixwith, whereand. [ABSTRACT FROM AUTHOR]

Abstract: Let and be bounded subsets and let and denote the generalized spectral radius of and the max algebra version of the generalized spectral radius of , respectively. We apply a single matrix description of to give a new elementary and straightforward proof of the Berger–Wang formula in max algebra and consequently a new short proof of the original Berger–Wang formula in the case of bounded subsets of non-negative matrices. We also obtain a new description of in terms of the Schur–Hadamard product and prove new trace and max-trace descriptions of and . In particular, we show that and where and . [Copyright &y& Elsevier]

Abstract: We establish a useful inequality for matrix power means of positive definite matrices. It eventually provides not only a new characterization of the Karcher mean (Cartan centroid, Riemannian geometric mean) but also a simple proof of the convergence of power means to the Karcher mean. Some new matrix mean inequalities on the Karcher and power means are presented. [Copyright &y& Elsevier]