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The stability and inertia of sign pattern matrices with entries in { + , − , 0 } associated with dynamical systems of second-order ordinary differential equations x ¨ = A x ˙ + B x are studied, where A and B are real matrices of order n. An equivalent system of first-order differential equations has coefficient matrix C = [ A B I O ] of order 2n, and eigenvalue properties are considered for sign patterns C = [ A B D O ] of order 2n, where A , B are the sign patterns of A , B respectively, and D is a positive diagonal sign pattern. For given sign patterns A and B where one of them is a negative diagonal sign pattern, results are determined concerning the potential stability and sign stability of C , as well as the refined inertia of C . Applications include the stability of such dynamical systems in which only the signs rather than the magnitudes of entries of the matrices A and B are known.

A sign pattern requires a unique inertia if every real matrix in the sign pattern class has the same inertia. Several sufficient or necessary conditions are given for a sign pattern to require a unique inertia. It is proved that a sign pattern requires a unique inertia if and only if it requires a unique refined inertia. All sign patterns of orders 2 and 3 that require a unique inertia are characterized. If the underlying graph of a sign pattern is a tree, then it is shown that any skew-symmetric sign pattern requires a unique inertia, and a linear-time algorithm is given to determine whether or not a symmetric sign pattern requires a unique inertia.

A set H n ⁎ of refined inertias for zero–nonzero patterns is introduced that is analogous to the set H n previously considered for sign patterns. For n = 3 and 4, a complete characterization of irreducible zero–nonzero patterns that allow or require H n ⁎ is given, and each zero–nonzero pattern that allows H n ⁎ has a signing that allows H n . In contrast, for n ≥ 5 a family of irreducible zero–nonzero patterns is given that allows H n ⁎ but for which no one signing allows H n .

The 18 non-isomorphic strongly connected orientations of the Petersen graph give rise to matrix patterns in which nonzero entries can be taken to be strictly positive, of arbitrary sign, or of fixed sign. The allowed refined inertias, in which the number of zero eigenvalues is split from others on the imaginary axis, are considered for each such pattern. Each nonnegative pattern is shown to have unique refined inertia determined by the number of required zero eigenvalues. For zero–nonzero patterns, a complete list of allowed refined inertias is given for each orientation. One particular sign pattern is presented that allows only two distinct refined inertias out of a possible 161 for a sign pattern of order 10.

For a real n × n matrix A having n + ( n − ) eigenvalues with positive (resp. negative) real part, n z zero eigenvalues and 2 n p nonzero pure imaginary eigenvalues, the refined inertia of A is ri ( A ) = ( n + , n − , n z , 2 n p ) . When n = 3 , let H 3 = { ( 0 , 3 , 0 , 0 ) , ( 0 , 1 , 0 , 2 ) , ( 2 , 1 , 0 , 0 ) } . A 3 × 3 sign pattern A requires refined inertia H 3 if { ri ( A ) | A has sign pattern A } = H 3 . Necessary and sufficient conditions for an irreducible sign pattern to require H 3 are given, and used to determine all such sign patterns (up to equivalence). These remove equivalences and complete the list of patterns given in [1, Appendix A] .

The problem of determining whether or not an n × n integer matrix is diagonally similar to an integer matrix with n − 1 off-diagonal entries equal to 1 is studied. Such a matrix is called integrally normalizable, and a zero–nonzero pattern is integrally normalizable if each matrix with this zero–nonzero pattern is integrally normalizable with respect to the same set of n − 1 off-diagonal entries. Matrices that are integrally normalizable with respect to a fixed spanning tree, and integrally normalizable zero–nonzero patterns are characterized. The maximum number of nonzero entries in an n × n integrally normalizable zero–nonzero pattern is shown to be n 2 + 3 n − 2 2 . Extensions to matrices over other integral domains are also presented.

The refined inertia ri ( A ) of a real n × n matrix A is the ordered 4-tuple ( n + , n - , n z , 2 n p ) where n + (resp. n - , n z , 2 n p ) is the number of positive (resp. negative, zero, nonzero pure imaginary) eigenvalues of A . Let H n = { ( 0 , n , 0 , 0 ) , ( 0 , n - 2 , 0 , 2 ) , ( 2 , n - 2 , 0 , 0 ) } . An n × n sign pattern S n requires H n if H n = { ri ( A ) | A has sign pattern S n } and allows H n if H n ∈ { ri ( A ) | A has sign pattern S n } . Sign patterns that require or allow H n are investigated for small values of n and for patterns with all diagonal entries negative. Examples are given relating these concepts to Hopf bifurcation in dynamical systems.

A square sign pattern A is potentially stable (PS) if there exists a real matrix having the sign pattern A and with all its eigenvalues having negative real parts. The characterization of PS sign patterns remains a long standing open problem. Here three techniques are given for the construction of larger order PS sign patterns from given PS sign patterns. These techniques are: construction of a sign pattern with a nested sequence of properly signed principal minors (a nest), bordering of a PS sign pattern, and use of a similarity transformation. The minimum number of nonzero entries in an irreducible PS sign pattern is determined for small orders and for an arbitrary sign pattern that allows a nest. For sign patterns of order at least four, a bordering construction leads to a new upper bound for this minimum number.

A matrix A ∈ M n ( R ) has a nest of positive principal minors if PAP T has positive leading principal minors for some permutation matrix P. Motivated by the fact that such a matrix A can be positively scaled so all its eigenvalues lie in the open right-half-plane, conditions are investigated so that a square sign pattern either requires or allows a nest of positive principal minors.

Given a vector u ∈ R 2 n , the principal minor assignment problem asks when is there an n × n matrix having its 2 n principal minors given by u . This paper explores the following related problem. Given a sequence r 0 r 1 ⋯ r n of 0s and 1s, does there exist an n × n real symmetric matrix that has a principal submatrix of rank k if and only if r k = 1 , for all 0 ⩽ k ⩽ n ? Certain conditions are shown to be necessary in order for this question to have an affirmative answer. Several families of matrices are constructed to attain certain classes of sequences. The problem is solved completely for n ⩽ 6 , and for 7 ⩽ n ⩽ 10 in the case of sequences beginning with 010.

A necessary and sufficient condition for the existence of the group inverse of a certain bipartite matrix is given, and an explicit formula is obtained for the group inverse in terms of its block submatrices. This form is used to derive a graph–theoretic description of the entries of the group inverse of some examples of such a matrix, including those corresponding to broom graphs. If the group inverse of a nonnegative matrix corresponding to a broom graph exists, then it is shown that this group inverse is signed. An open question about group inverses of more general bipartite matrices is posed and a summary of cases for which its answer is known is given.

An n × n sign pattern matrix has entries in { + , - , 0 } . This paper surveys the following problems concerning spectral properties of sign pattern matrices: sign patterns that allow all possible spectra (spectrally arbitrary sign patterns); sign patterns that allow all inertias (inertially arbitrary sign patterns); sign patterns that allow nilpotency (potentially nilpotent sign patterns); and sign patterns that allow stability (potentially stable sign patterns). Relationships between these four classes of sign patterns are given, and several open problems are identified.

A primitive digraph D on n vertices has large exponent if its exponent, γ ( D ) , satisfies α n ⩽ γ ( D ) ⩽ w n , where α n = ⌊ w n / 2 ⌋ + 2 and w n = ( n - 1 ) 2 + 1 . It is shown that the minimum number of arcs in a primitive digraph D on n ⩾ 5 vertices with exponent equal to α n is either n + 1 or n + 2 . Explicit constructions are given for fixed n even and odd, for a primitive digraph on n vertices with exponent α n and n + 2 arcs. These constructions extend to digraphs with some exponents between α n and w n . A necessary and sufficient condition is presented for the existence of a primitive digraph on n vertices with exponent α n and n + 1 arcs. Together with some number theoretic results, this gives an algorithm that determines for fixed n whether the minimum number of arcs is n + 1 or n + 2 .

An n by n sign pattern S is inertially arbitrary if each ordered triple (n1, n2, n3) of nonnegative integers with n1 + n2 + n3 = n is the inertia of some real matrix in Q ( S ) , the sign pattern class of S . If every real, monic polynomial of degree n having a positive coefficient of xn−2 is the characteristic polynomial of some matrix in Q ( S ) , then it is shown that S is inertially arbitrary. A new family of irreducible sign patterns G 2 k + 1 ( k ⩾ 2 ) is presented and proved to be inertially arbitrary, but not potentially nilpotent (and thus not spectrally arbitrary). The well-known Nilpotent-Jacobian method cannot be used to prove that G 2 k + 1 is inertially arbitrary, since G 2 k + 1 has no nilpotent realization. In order to prove that Q ( G 2 k + 1 ) allows each inertia with n3 ⩾ 1, a realization of G 2 k + 1 with only zero eigenvalues except for a conjugate pair of pure imaginary eigenvalues is identified and used with the Implicit Function Theorem. Matrices in Q ( G 2 k + 1 ) with inertias having n3 = 0 are constructed by a recursive procedure from those of lower order. Some properties of the coefficients of the characteristic polynomial of an arbitrary matrix having certain fixed inertias are derived, and are used to show that G 5 and G 7 are minimal inertially arbitrary sign patterns.

An n × n sign pattern S n is potentially nilpotent if there is a real matrix having sign pattern S n and characteristic polynomial x n . A new family of sign patterns C n with a cycle of every even length is introduced and shown to be potentially nilpotent by explicitly determining the entries of a nilpotent matrix with sign pattern C n . These nilpotent matrices are used together with a Jacobian argument to show that C n is spectrally arbitrary, i.e., there is a real matrix having sign pattern C n and characteristic polynomial x n + ∑ i = 1 n ( - 1 ) i μ i x n - i for any real μ i . Some results and a conjecture on minimality of these spectrally arbitrary sign patterns are given.

We characterize those square partial matrices whose specified entries constitute a rectangular submatrix that may be completed to an inverse M -matrix. Together with the notion of an interior inverse M -matrix, this is used to show that any positive matrix is a sum of inverse M -matrices and to estimate the number of summands needed to represent a given matrix. Nonnegative matrices are also considered. There are substantial differences from the analogous problem of decomposing a positive matrix as a sum of totally positive matrices. In particular, the upper bound on the number of inverse M -matrix summands is much less than that in the totally positive case (although an example is given to show that the number of totally positive summands may be less than the required number of inverse M -matrix summands).

The exponent γ(D) of a primitive digraph D is the smallest m such that for each ordered pair of not necessarily distinct vertices (u, v), there exists a u ⇝ v walk of length m. If λ(D) is the set of all cycle lengths, then the circumdiameter of D, denoted dc(λ(D)), is the maximum over all ordered pairs of not necessarily distinct vertices (u, v) of the length of a shortest u ⇝ v walk that intersects cycles of all lengths. It is well known that γ(D) ⩽ ϕ(λ(D)) + dc(λ(D)), in which ϕ(λ(D)) is the Frobenius–Schur index. Several new sufficient conditions and families of digraphs for which equality holds in the above upper bound are given, and some families of digraphs D for which γ(D) = ϕ(λ(D)) + dc(λ(D)) − 1 are defined. Additional sufficient conditions for equality in the above upper bound for γ(D) and a new upper bound for dc(λ(D)) are given for digraphs with large exponent, that is, digraphs on n vertices having γ ( D ) ⩾ ⌊ ( n - 1 ) 2 + 1 2 ⌋ + 2 . The circumdiameter and bounds on the exponent for the digraph of a Leslie matrix are found.

The Moore–Penrose inverse of a real matrix having no square submatrix with two or more diagonals is described in terms of bipartite graphs. For such a matrix, the sign of every entry of the Moore–Penrose inverse is shown to be determined uniquely by the signs of the matrix entries; i.e., the matrix has a signed generalized inverse. Necessary and sufficient conditions on an acyclic bipartite graph are given so that each nonnegative matrix with this graph has a nonnegative Moore–Penrose inverse. Nearly reducible matrices are proved to contain no submatrix having two or more diagonals, implying that a nearly reducible matrix has a signed generalized inverse. Furthermore, it is proved that the term rank and rank are equal for each submatrix of a nearly reducible matrix.

A maximal graph is a connected graph with degree sequence not majorized by the degree sequence of any other graph. Relationships between three different sequences of integers that describe maximal graphs are given. Recursive and explicit forms of the characteristic polynomial of the adjacency matrix of a maximal graph are derived. A correspondence between maximal graphs and connected graphs with stepwise adjacency matrices shows that, among all connected graphs with n vertices and e edges, the graph with maximal spectral radius is a maximal graph. Such maximal graphs are identified for certain values of n and e.

Let A be an adjacency matrix of a tree T with n vertices. Conditions are determined for the existence of a fixed permutation matrix P that maximizes the quadratic form x t P t APx over all nonnegative vectors x with entries arranged in nondecreasing order. This quadratic form problem is completely solved, and its answer leads to a corresponding solution for the problem of determining conditions for the existence of a fixed permutation matrix P that maximizes the largest eigenvalue of matrices of the form PDP t +A , over all real diagonal matrices D with nondecreasing diagonal entries. It is shown that there is a tree with six vertices for which neither of the problems has a solution, and all other trees with six or fewer vertices have solutions for both problems. By duality, the results also apply to the analogous problem of minimizing the smallest eigenvalue of matrices of the form PDP t +A .

A family of n×n symmetric circulant (0, 1) matrices is studied. It is shown that the determinant of each matrix is (−1) n−1 (n−1) , a property shared with the adjacency matrix of the complete graph on n nodes. As a result, each matrix in this family generates an incomplete graph that forms a counterexample to a recent conjecture.

An n×n sign pattern matrix S is an inertially arbitrary pattern (IAP) if each nonnegative triple (n 1 ,n 2 ,n 3 ) with n 1 +n 2 +n 3 =n is the inertia of a matrix with sign pattern S . Analogously, S is a spectrally arbitrary pattern (SAP) if, for any given real monic polynomial r(x) of order n , there is a matrix with sign pattern S and characteristic polynomial r(x) . Focusing on tree sign patterns, consider the n×n tridiagonal sign pattern T n with each superdiagonal entry positive, each subdiagonal entry negative, the (1,1) entry negative, the (n,n) entry positive, and every other entry zero. It is conjectured that T n is an IAP. By constructing matrices A n with pattern T n , it is proved that T n allows any inertia with n 3 ∈{0,1,2,n−1,n} for all n⩾2 . This leads to a proof of the conjecture for n⩽5 . The truth of the conjecture is extended to n⩽7 by showing the stronger result that T n is a SAP. The proof of this latter statement involves finding a matrix A n with pattern T n that is nilpotent. Further questions about patterns that are SAPs and IAPs are considered.

An elementary bidiagonal (EB) matrix has every main diagonal entry equal to 1, and exactly one off-diagonal nonzero entry that is either on the sub- or super-diagonal. If matrix A can be written as a product of EB matrices and at most one diagonal matrix, then this product is an EB factorization of A. Every matrix is shown to have an EB factorization, and this is related to LU factorization and Neville elimination. The minimum number of EB factors needed for various classes of n-by-n matrices is considered. Some exact values for low dimensions and some bounds for general n are proved; improved bounds are conjectured. Generic factorizations that correspond to different orderings of the EB factors are briefly considered.

Two different generalizations of the Perron—Frobenius theory to the matrix pencil Ax = λBx are discussed, and their relationships are studied. In one generalization, which was motivated by economics, the main assumption is that (B − A)−1A is nonnegative. In the second generalization, the main assumption is that there exists a matrix X ⩾ 0 such that A = BX. The equivalence of these two assumptions when B is nonsingular is considered. For ρ(|B−1A|) < 1, a complete characterization, involving a condition on the di-graph of B−1A, is proved. It is conjectured that the characterization holds for ρ(B−1A) < 1, and partial results are given for this case.

Sufficient conditions are determined for an n - by - n zero-nonzero pattern to allow a nested sequence of nonzero principal minors. A method is then given to sign such a pattern so that it allows a nested sequence of k - by - k principal minors with sign (−1) k for 1 ⩽ k ⩽ n . It is a classical result that a sign pattern with such a sequence is potentially stable; thus classes of potentially stable sign patterns can be identified. A necessary and sufficient condition for a certain tree sign-pattern class to be potentially stable is also given.

A nearly sign-nonsingular (NSNS) matrix is a real n × n matrix having at least two nonzero terms in the expansion of its determinant with precisely one of these terms having opposite sign to all the other terms. Using graph-theoretic techniques, we study the spectra of irreducible NSNS matrices in normal form. Specifically, we show that such a matrix can have at most one nonnegative eigenvalue, and can have no nonreal eigenvalue z in the sector { z: | arg z| ⩽ κ (n − 1) }. We also derive results concerning the sign pattern of inverses of these matrices.

An n-by-n matrix Bn is sign nonsingular (SNS) if every matrix with the same sign pattern as Bn is nonsingular. A given SNS matrix determines an equivalence class (with respect to transposition and multiplication by permutation and signature matrices) of SNS matrices, all of which have the same number of zero entries. Such a matrix is maximal if no zero entry can be set nonzero so that the resulting matrix is SNS, and is fully indecomposable if it does not have an (n − k)-by-k zero submatrix for some k, where 1 ⩽ k ⩽ n − 1. For fixed n, the Hessenberg matrix is known to represent the unique equivalence class with the minimum number of zero entries, namely ( n−1 2 ). We prove that for n ⩾ 5, there is exactly one equivalence class of fully indecomposable maximal SNS matrices with ( n−1 2 ) + 1 zero entries. Similarly, for n ⩾ 5, we prove that there are exactly two such equivalence classes having ( n−1 2 ) + 2 zero entries. For these proofs, we identify two new infinite classes of fully indecomposable maximal SNS matrices, which can be obtained by stretching known SNS matrices.

Let A be an n-by-n irreducible, entrywise nonnegative matrix. For a given t > 0, we consider the problem of maximizing the Perron root of a nonnegative, diagonal, trace t perturbation of A. Because of the convexity of the Perron root as a function of diagonal entries, the maximum occurs for some tEii. Such an index i, which is called a winner, may depend on t. We show how to determine the (nonempty) set of indices i that are winners for all sufficiently small t and the possibly different (nonempty) set of indices that are winners for all sufficiently large t. We also show how to determine if there are indices that are winners for all t.

For an essentially nonnegative matrix A , we consider the problem of minimizing the dominant eigenvalue of A + D over real diagonal matrices D with zero trace. The solution is closely related to the unique line-sum-symmetric diagonal similarity of A in the irreducible case, and we describe the solution for general essentially nonnegative A . The minimizer D is always unique, and we characterize those matrices A for which the minimizer D is 0. We solve the problem for several classes of matrices by finding the line-sum-symmetric diagonal similarity as an explicit function of the entries of A in some cases, and in terms of the zeros of polynomials with coefficients constructed from the entries of A in others.

The class of real matrices which are both monotone (inverse positive) and positive stable is investigated. Such matrices, called N -matrices, have the well-known class of nonsingular M -matrices as a proper subset. Relationships between the classes of N -matrices, M -matrices, nonsingular totally nonnegative matrices, and oscillatory matrices are developed. Conditions are given for some classes of matrices, including tridiagonal and some Toeplitz matrices, to be N -matrices.

Let A be an arbitrary n×n matrix, partitioned so that if A=[Aij], then all submatrices Aii are square. If x is a positive vector, it is well-known that G( x ) =∪ N i=1 G i ( x ) , where G i ( x ) = z ‖(zI − A ii ) −1 ‖ −1 ⩽ 1 x i ∑ j = 1 j ≠ i N `‖A ij ‖x j , contains all the eigenvalues of A. The purpose of this paper is to give a new definition of the concept of an isolated subregion of G(x). An algorithm is given for obtaining the best such isolated subregion in a certain sense, and examples are given to show that tighter bounds for some eigenvalues of A may be obtained than with previous algorithms. For ease of computation, each subregion Gi(x) is replaced by the union of circular disks centered at the eigenvalues of Aii.

We show that if A is an M -matrix for which the length of the longest simple cycle in its associated undirected graph G ( A ) is at most 3, then every minor of A has determined sign (nonnegative or nonpositive), independent of the magnitudes of the matrix entries. Consequently, if A and B are M -matrices such that G ( A ) and G ( B ) are subgraphs of an undirected graph with longest simple cycle at most 3, then all principal minors of AB are nonnegative.

Let ρ i ( t ) = ρ ( A + tE ii ) denote the spectral radius of the sum of an irreducible nonnegative matrix A and a matrix tE ii that has a single nonzero entry, namely t > 0 in the i , i position. We consider qualitative aspects of maximizing ρ i ( t ), especially identifying maximizing indices i , and indices i and j that tie [i.e., ρ i ( t ) = ρ j ( t ) for all t > 0]. If the digraph of A is a directed cycle, then all vertices tie; whereas if the digraph of A is a star, then the center is the unique maximizing vertex. When A is the (0, 1) adjacency matrix of a graph, we give sufficient conditions in terms of the orbits of vertices for a tie. For complete bipartite graphs and for paths, vertices i are identified that maximize ρ i ( t ) for all t > 0. However, even for a tree, it is not in general true that some fixed vertex i maximizes ρ i ( t ) for all t > 0.

Let S be a subset of diagonal entries of an n × n complex matrix A . When the members of S have a common value which is equal to an eigenvalue of A , then S is a critical diagonal set of A . The existence of such a set is equivalent to the matrix A, obtained from A by setting its diagonal entries equal to zero, having an s × t zero submatrix with s + t ⩾ n + 1. If S is a minimal critical diagonal set, then equality holds. A zero submatrix of A is used to identify the elements in a critical diagonal set. Then a combinatorial approach is taken in order to study the eigenspace of an eigenvalue associated with a critical diagonal set and to make observations regarding the Frobenius normal form of A and reducibility of A .

The n-by-n tridiagonal sign pattern T-n has every superdiagonal entry positive, every sub-diagonal entry negative, the (1, 1) entry negative, the (n, n) entry positive and every other diagonal entry zero. Inertia and spectral results for matrices An having the sign pattern Tn are proved using new techniques on low rank perturbations. It is also shown (by using MAPLE) that for 8 less than or equal to n less than or equal to 16, T-n allows any spectrum. These results extend those previously in the literature, and strengthen the conjecture that T-n allows any spectrum for all values of n. In addition, bounds on the algebraic multiplicity of an eigenvalue of a low rank perturbation of a general matrix are obtained. (C) 2003 Elsevier Inc. All rights reserved.

We characterize matrices A ϵ C n × n whose zero-nonzero pattern requires that the controllability matrix [ b Ab A 2 b ⋯ A n − 1 b ] ϵ C n × n be of full rank, where b ϵ C n × 1 has exactly one nonzero entry. When all the diagonal entries of A are nonzero, we show that this occurs if and only if QAQ T is unreduced upper Hessenberg, with Q a permutation matrix for which Qb = [ b 1 , 0,…, 0] T . We also characterize matrices A whose zero-nonzero pattern requires that the controllability matrix be of deficient rank.

It is well known that if all the elementary symmetric functions of the eigenvalues of an n -by- n matrix are positive, then all its eigenvalues lie in the region of the complex plane { z :-π+π/ n z n }. Let A denote an n -by- n matrix with all diagonal entries nonzero and for which the length of the longest cycle in its directed graph is k ,2⩽ k ⩽ n . If, in addition, all the cycles in the directed graph of - A are signed negatively, then the elementary symmetric functions of the eigenvalues of A are positive and we ask whether its eigenvalues lie in the region { z :-π+π/ k z k }. This is known to be true when k =2 (sign stability), and we prove it here for k = n -1. We provide a counterexample for the case k = n -3 and discuss related questions for more general classes of matrices with restricted length of the longest cycle.

It is shown that an arbitrary m × n positive matrix can be written as a sum of at most min { m , n } totally positive matrices, and that this is in general the best possible value for the number of summands. Sufficient conditions are given under which fewer than min { m , n } totally positive summands are needed.

Results are given concerning the LU factorization of H-matrices, and Gaussian elimination with column-diagonal-dominant pivoting is shown to be applicable to H-matrices. This algorithm, which uses a symmetric permutation to exchange the most diagonally dominant column of the unreduced submatrix into the pivotal position, is shown to be numerically stable by deriving an upper bound on the growth factor associated with the backward error analysis for Gaussian elimination.

For an n -by- n P -matrix A = [ a ij ] having LU factorization A = LU with U = [ u ij ], we determine combinatorial circumstances for which u ij is unambiguously signed for a given pair i ≤ j , or all such pairs, and give corresponding results for L . A qualitative P -matrix is a sign nonsingular matrix with all diagonal entries positive. For such a matrix A , we give additional sufficient conditions for an entry u ij or the matrix U to be unambiguous. If A is a qualitative P -matrix with A −1 unambiguously signed, we prove that the matrices L , U , L −1 , and U −1 are all unambiguous.

The notions of irreducibility, primitivity, and exponent of a nonegative matrix are generalized to k -tuples of non-negative matrices of the same order. It is shown that for each positive integer k , the maximum exponent of a primitive k -tuple of n by n nonnegative matrices is Θ ( n k +1 ).

The generalized Gerschgorin disks of a partitioned matrix A , as introduced by D. G. Feingold and R. S. Varga ( Pacif. J. Math. 12 1241–1250 (1962)), are shown to give the smallest region covering all eigenvalues of all matrices from a certain class related to A .