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The paper considers the classes of GSDD1, GSDD 1 ∗ , and SD-SDD matrices, which contain the class of SDD (strictly diagonally dominant) matrices and are contained in the class of nonsingular H -matrices. New upper bounds on ‖ A - 1 ‖ ∞ for GSDD1, GSDD 1 ∗ , and SD-SDD matrices A, generalizing known upper bounds for S-SDD, SDD 1 ∗ , and GSDD1 matrices, are established and compared. [ABSTRACT FROM AUTHOR]

The paper suggests two general approaches to deriving upper bounds for ‖ A - 1 ‖ ∞ for some eventually H -matrices A. The approaches are illustrated by considering the classes of eventually SDD and eventually DSDD matrices, for which improvements of the known bounds are obtained. Also it is indicated that the approaches suggested are actually applicable to considerably larger matrix classes. [ABSTRACT FROM AUTHOR]

The paper suggests and illustrates a simple unified approach to deriving upper bounds for the dominant eigenvalues of the so-called PF matrices (or matrices with the Perron–Frobenius property) from those for the Perron root of a nonnegative matrix. [ABSTRACT FROM AUTHOR]

The paper continues the study of the recently introduced class of SDD1 matrices. The class of general SDD1 matrices and its three subclasses are considered. In particular, it is shown that SDD1 matrices are nonsingular ℌ-matrices. Also parameter-free upper bounds for the l∞-norm of the inverses to SDD1 matrices are derived. The block triangular form to which any SDD1 matrix can be brought by a symmetric permutation of its rows and columns is described. [ABSTRACT FROM AUTHOR]

The paper suggests a general approach to deriving upper bounds for ‖A−1Q‖∞ from those for ‖A−1‖∞ for matrices A belonging to different subclasses of the class of nonsingular ℌ-matrices. The approach is applied to SDD, S-SDD, OBS, OB, and Nekrasov matrices. [ABSTRACT FROM AUTHOR]

The paper continues the study of block generalizations of Nekrasov matrices and introduces two new classes of the so-called G ˜ N and BJN matrices and compares them with the previously introduced class of GN matrices. Different properties of G ˜ N and BJN matrices are established. In particular, it is proved that the classes G ˜ N and {BJN} are closed with respect to Schur complements and monotone with respect to block partitioning. Also upper bounds for the norms ‖A−1‖∞ of inverses of GN, G ˜ N , and BJN matrices A are considered. General results obtained are specialized to the case of block two-by-two matrices with scalar first diagonal block. [ABSTRACT FROM AUTHOR]

New general upper and lower bounds for the Perron root of a nonnegative matrix, which involve nonempty proper subsets of the index set and the matrix sparsity pattern, are suggested, and some special cases are considered. Also the nonsingularity criteria related to the upper bounds presented, which generalize some known results on subclasses of nonsingular ℋ-matrices, are derived. [ABSTRACT FROM AUTHOR]

The paper introduces the so-called generalized Nekrasov (GN) matrices, which provide a block extension of the ordinary Nekrasov matrices. Basic properties of GN matrices are studied. In particular, it is proved that the GN matrices form a subclass of nonsingular H-matrices and this subclass is closed with respect to Schur complements obtained by eliminating leading principal block submatrices. Also an upper bound for the l∞-norm of the inverse to a GN matrix, which generalizes the known bound for Nekrasov matrices, is obtained. The case of block two-by-two GN matrices with scalar first diagonal blocks, which prove to be Dashnic–Zusmanovich matrices of the first type, is considered separately. The bounds obtained are applied to SDD matrices. [ABSTRACT FROM AUTHOR]

The paper introduces new classes of nonsingular matrices, which contain some known subclasses of the class of nonsingular H-matrices, such as the Nekrasov, Q-Nekrasov, {P1, P2}-Nekrasov, and DZ matrices. For matrices in the classes introduced, upper bounds for ‖A−1‖∞ are derived (in a unified manner) and shown to improve the known bounds for matrices from the corresponding subclasses of the nonsingular H-matrices. Bibliography: 20 titles. [ABSTRACT FROM AUTHOR]

The paper considers the so-called P-Nekrasov and {P1, P2}-Nekrasov matrices, defined in terms of permutation matrices P, P1, P2, which generalize the well-known notion of Nekrasov matrices. For such matrices A, available upper bounds on ‖A−1‖∞ are recalled, and new upper bounds for the P-Nekrasov and {P1, P2}-Nekrasov matrices are suggested. It is shown that the latter bound generally improves the earlier bounds, as well as the bound for the inverse of a P-Nekrasov matrix and the classical bound for the inverse of a strictly diagonally dominant matrix. Bibliography: 12 titles. [ABSTRACT FROM AUTHOR]

The paper considers some subclasses of the class of nonsingular H-matrices whose definitions involve matrix sparsity pattern. For matrices A in these subclasses, upper bounds for ‖A−1‖∞ are derived and shown to be sharper than the corresponding bounds ignoring matrix sparsity. Bibliography: 26 titles. [ABSTRACT FROM AUTHOR]

The paper is mainly devoted to studying the so-called Dashnic–Zusmanovich type (DZT) matrices, introduced recently. Interrelations among the DZT matrices and related subclasses of the class of nonsingular H -matrices, namely, the Dashnic–Zusmanovich (DZ) and S-SDD matrices are considered. Upper bounds for the l∞-norms of the inverses to DZT, DZ, and strictly diagonally dominant (SDD) matrices are obtained. A new eigenvalue inclusion set is provided. [ABSTRACT FROM AUTHOR]

The paper presents new nonsingularity conditions for n ×n matrices, which involve a subset S of the index set {1,..., n} and take into consideration the matrix sparsity pattern. It is shown that the matrices satisfying these conditions form a subclass of the class of nonsingular H -matrices, which contains some known matrix classes such as the class of doubly strictly diagonally dominant (DSDD) matrices and the class of Dashnic–Zusmanovich type (DZT) matrices. The nonsingularity conditions established are used to obtain the corresponding eigenvalue inclusion sets, which, in their turn, are used in deriving new inclusion sets for the singular values of a square matrix, improving some recently suggested ones. [ABSTRACT FROM AUTHOR]

The new upper boundλmaxA≤∑k=1p+1i≡kmodp+1maxλmaxAiifor the largest eigenvalue of a Hermitian positive semidefinite block banded matrix A = (Aij ) of block semibandwidth p is suggested. In the special case where the diagonal blocks of A are identity matrices, the latter bound reduces to the boundλmaxA≤p+1,depending on p only, which improves the bounds established for such matrices earlier and extends the boundλmaxA≤2,old known for p = 1, i.e., for block tridiagonal matrices, to the general case p ≥ 1. [ABSTRACT FROM AUTHOR]

The paper suggests a general approach to deriving upper bounds for the spectral radii of weighted digraphs. The approach is based on the generalized Wielandt lemma (GWL), which reduces the problem of bounding the spectral radius of a given block matrix to bounding the Perron root of the matrix composed of the norms of the blocks. In the case of the adjacency matrix of weighted graphs and digraphs where all the blocks are square positive (semi)definite matrices of the same order, the GWL takes an especially nice simple form. The second component of the approach consists in applying known upper bounds for the Perron root of a nonnegative matrix. It is shown that the approach suggested covers, in particular, the known upper bounds of the spectral radius and allows one to describe the equality cases. [ABSTRACT FROM AUTHOR]

The paper introduces new subclasses, called P $$ \mathrm{\mathscr{H}} $$ N( π) and P $$ \mathrm{\mathscr{H}} $$ QN( π), of (nonsingular) $$ \mathrm{\mathscr{H}} $$ -matrices of order n dependent on a partition π of the index set {1, . . ., n}, which generalize the classes P $$ \mathrm{\mathscr{H}} $$ ( π), introduced previously, and contain, in particular, such subclasses as those of strictly diagonally dominant (SDD), Nekrasov, S-SDD, S-Nekrasov, QN, and P $$ \mathrm{\mathscr{H}} $$ ( π) matrices. Properties of the matrices introduced are studied, and upper bounds on their inverses in l norm are obtained. Block generalizations of the classes P $$ \mathrm{\mathscr{H}} $$ N( π) and P $$ \mathrm{\mathscr{H}} $$ QN( π) in the sense of Robert are considered. Also a general approach to defining subclasses $$ {\mathcal{K}}^{\pi } $$ of the class $$ \mathrm{\mathscr{H}} $$ containing a given subclass $$ \mathcal{K} $$ ⊂ $$ \mathrm{\mathscr{H}} $$ and dependent on a partition π is presented. [ABSTRACT FROM AUTHOR]

The paper considers some classes of block matrices that can be regarded as block generalizations of the class of $$ \mathrm{\mathscr{H}} $$ -matrices and interrelations among them. New bounds for the infinity norm of inverses for matrices in the classes under consideration are suggested. [ABSTRACT FROM AUTHOR]

The paper suggests generalizations of some known sufficient nonsingularity conditions for matrices with constant principal diagonal and the corresponding eigenvalue inclusion sets to the cases of arbitrary matrices and matrices with nonzero diagonal entries. Bibliography: 11 titles. [ABSTRACT FROM AUTHOR]

The paper suggests upper bounds for the l norm of the inverses to block matrices belonging to certain subclasses of the class of block $$ \mathrm{\mathscr{H}} $$-matrices, which improve and supplement known results. [ABSTRACT FROM AUTHOR]

Two-sided bounds on | det A| for Nekrasov and S-Nekrasov matrices A are obtained. It is shown that for Nekrasov matrices the new bounds improve the known bounds of Bailey and Crabtree. As to the relatively recently introduced S-Nekrasov matrices, no bounds on their determinants have been suggested, as far as the author is aware. Bibliography: 13 titles. [ABSTRACT FROM AUTHOR]

It is known that the Nekrasov and S-Nekrasov matrices form subclasses of (nonsingular) H-matrices. The paper presents some necessary and sufficient conditions for a square matrix with complex entries to be a Nekrasov and an S-Nekrasov matrix. In particular, characterizations of the Nekrasov and S-Nekrasov matrices in terms of the diagonal column scaling matrices transforming them into strictly diagonally dominant matrices are obtained. Bibliography: 15 titles. [ABSTRACT FROM AUTHOR]

The paper considers upper bounds for the infinity norm of the inverse for matrices in two subclasses of the class of (nonsingular) H-matrices, both of which contain the class of Nekrasov matrices. The first one has been introduced recently and consists of the so-called S-Nekrasov matrices. For S-Nekrasov matrices, the known bounds are improved. The second subclass consists of the socalled QN- (quasi-Nekrasov) matrices, which are defined in the present paper. For QN-matrices, an upper bound on the infinity norm of the inverses is established. It is shown that in application to Nekrasov matrices the new bounds are generally better than the known ones. Bibliography: 15 titles. [ABSTRACT FROM AUTHOR]

In the paper, some inequalities interrelating the singular values of a square matrix A with complex entries with the Perron roots of the associated nonnegative matrices $ {P_A}=\left| {A\left| \circ \right|\left. A \right|} \right.\ and\ \left( {{P_A}+P_A^T} \right)/2 $ are derived. The results obtained are applied to (0, 1)-matrices. [ABSTRACT FROM AUTHOR]

In the paper, a new upper bound for the largest eigenvalue q1 of the signless Laplacian Q = D + A of a graph G, generalizing and improving the known bound q ≤ Δ + Δ, where Δ ≥ ・・・ ≥ Δ are the ordered vertex degrees, and also new lower bounds for the second largest eigenvalue q of Q are proved. As implications, upper bounds for the difference q − μ of the largest eigenvalues of Q and of the Laplacian matrix L = D − A, an upper bound for the largest eigenvalue of the adjacency matrix AG, and an upper bound for the difference q − q are obtained. All the bounds suggested are expressed in terms of the vertex degrees. [ABSTRACT FROM AUTHOR]

A new upper bound on the infinity norm of the inverse to a Nekrasov matrix, improving two recent bounds by L. Cvetkovi'c et al. (Appl. Math. Comput., 219, 5020-5024 (2013)), is obtained. Also it is shown that for a strictly diagonally dominant matrix, the bound suggested improves the classical Varah upper bound. [ABSTRACT FROM AUTHOR]

The paper suggests uppe bound on the second largest eigenvalue and the sum of two largest eigenvalues of symmetric nonnegative matrices and graphs. Conditions necessary and sufficient for some of the counds to be attained are established. Special attention is paid to the subclass of matrices with zero diagonal entries and with off-diagonal entries not exceeding unity, which obviously contains the adjacency matrices of undirected graphs. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

The paper suggests a new approach to deriving lower bounds for the Laplacian spectral radius and upper bounds for the smallest eigenvalve of the signless Laplacian of an undirected simple r-partite graph on n vertices, 2 ≤ r ≤ n. The approach is based on inequalities for the extreme eigenvalves of a block-partitioned Hermitian matrix, established earlier, and on the Rayleigh principle. Specific lower and upper bounds generalizing known results and extending them from r = 2 to r ≥ 2 are considered, and the cases where these bounds are sharp are described. Bibliography: 13 titles. [ABSTRACT FROM AUTHOR]

The paper suggests new two-sided bounds for the Perron root of a weakly irreducible nonnegative matrix, which depend on the circuits of length no less than two in the associated directed graph and, in some cases, improve Al'pin's bounds, suggested in 1995. Two approaches to deriving circuit bounds are considered. Bibliography: 5 titles. [ABSTRACT FROM AUTHOR]

Let A = D + B be a block r × r, r ≥ 2, Hermitian matrix of order n, where D is the block diagonal part of A. The main results of the paper are Theorems 2.1 and 2.2, which state the sharp inequalities and analyze the equality cases. Some implications of these results are indicated. As applications, matrices occurring in spectral graph theory are considered, and new lower bounds on the chromatic number of a graph are obtained. Bibliography: 7 titles. [ABSTRACT FROM AUTHOR]

The paper suggests sufficient nonsingularity conditions for matrices in terms of certain determinantal relations of diagonal dominance type, which improve and generalize some known results. These conditions are used to describe new eigenvalue inclusion sets and to derive new two-sided bounds on the determinants of matrices satisfying them. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

The paper refines the classical Ostrowski disk theorem and suggests lower bounds for the smallest-in-modulus eigenvalue and the smallest singular value of a square matrix under certain diagonal dominance conditions. A lower bound for the smallest-in-modulus eigenvalue of a product of m ≥ 2 matrices satisfying joint diagonal dominance conditions is obtained. The particular cases of the bounds suggested that correspond to the infinity norm are discussed separately and compared with some known results. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

The paper presents new bounds for the inverses of the so-called PM- and PH-matrices. Also bounds for the spectral radii of the inverses to PM- and PH-matrices are obtained, and the monotonicity of these bounds with respect to the underlying partition of the index set is established. Finally, the so-called quasi- PM- and quasi- PH-matrices are introduced, and bounds for the inverses of such matrices are suggested. Bibliography: 10 titles. [ABSTRACT FROM AUTHOR]

New characterizations of the so-called PM-and PH-matrices in terms of generalized diagonal dominance are obtained and used to derive some of their properties. Also it is proved that a PH-matrix necessarily has a certain number of strictly diagonally dominant rows. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

The paper considers different circuit inclusion sets for the singular values of a square matrix. It is shown that in both the general and the structurally symmetric cases, the circuit inclusion sets are of theoretical interest only, because they coincide with the much simpler Ostrowski–Brauer type inclusion sets, which should be used in practice. Bibliography: 10 titles. [ABSTRACT FROM AUTHOR]

The paper presents a general approach to deriving inclusion sets for the singular values of a matrix A = (aij) ∈ ℂn×n. The key to the approach is the following result: If σ is a singular value of A, then a certain matrix C(σ, A) of order 2n, whose diagonal entries are σ2 − | aii|2, i = 1, ..., n, is singular. Based on this result, we use known diagonal-dominance type nonsingularity conditions to obtain inclusion sets for the singular values of A. Scaled versions of the inclusion sets, allowing one, in particular, to obtain Ky Fan type results for the singular values, are derived by passing to the conjugated matrix D−1C(σ, A)D, where D is a positive-definite diagonal matrix. Bibliography: 16 titles. [ABSTRACT FROM AUTHOR]

The paper generalizes certain inclusion sets for the singular values of a square matrix to the case of an m × n matrix. In particular, it is shown that under a nonrestrictive assumption on the ordering of the matrix columns (if m < n) or the matrix rows (if m > n), a natural counterpart of the Gerschgorin theorem on the eigenvalue location is valid. Bibliography: 14 titles. [ABSTRACT FROM AUTHOR]

The paper presents new two-sided bounds for the Perron root of a block-partitioned nonnegative matrix, improving Chistyakov’s bounds. The equality cases are analyzed. As an application, new conditions sufficient for a complex matrix to be a nonsingular H-matrix are obtained. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

The paper studies a subclass, referred to as PBDD(n1, n2), of the class of nonsingular H-matrices. A new characterization of matrices in PBDD(n1, n2) is suggested. Two-sided bounds for the determinants of matrices in the class PBDD(n1, n2) are derived, and their applications to strictly diagonally dominant matrices and to matrices with the Ostrowski-Brauer diagonal dominance are presented. An upper bound for the infinity norms of the inverses of matrices in PBDD(n1, n2) is considered. Extensions to the case of block k × k matrices, k ≥ 2, are addressed. Bibliography: 17 titles. [ABSTRACT FROM AUTHOR]

The paper presents a new monotonicity property of the Perron root of a nonnegative matrix. It is shown that this new property implies known monotonicity properties and also the Chistyakov two-sided bounds for the Perron root of a block-partitioned nonnegative matrix. Moreover, based on the monotonicity property suggested, the equality cases in Chistyakov’s theorem are analyzed. Applications to bounding above the spectral radius of a complex matrix are presented. Bibliography: 9 titles. [ABSTRACT FROM AUTHOR]

Given a finite set {Ax}x ∈ X of nonnegative matrices, we derive joint upper and lower bounds for the row sums of the matrices D−1 A(x) D, x ∈ X, where D is a specially chosen nonsingular diagonal matrix. These bounds, depending only on the sparsity patterns of the matrices A(x) and their row sums, are used to obtain joint two-sided bounds for the Perron roots of given nonnegative matrices, joint upper bounds for the spectral radii of given complex matrices, bounds for the joint and lower spectral radii of a matrix set, and conditions sufficient for all convex combinations of given matrices to be Schur stable. Bibliography: 20 titles. [ABSTRACT FROM AUTHOR]

The paper presents new diagonal dominance type nonsingularity conditions for n × n matrices formulated in terms of circuits of length not exceeding a fixed number r ≥ 0 and simple paths of length r in the digraph of the matrix. These conditions are intermediate between the diagonal dominance conditions in terms of all paths of length r and Brualdi’s diagonal dominance conditions, involving all the circuits. For r = 0, the new conditions reduce to the standard row diagonal dominance conditions $$\left| {a_{ii} } \right| \geqslant \sum\limits_{j \ne i} {\left| {a_{ij} } \right|} $$ , i = 1, ..., n, whereas for r = n they coincide with the Brualdi circuit conditions. Thus, they connect the classical Lévy-Desplanques theorem and the Brualdi theorem, yielding a family of sufficient nonsingularity conditions. Further, for irreducible matrices satisfying the new diagonal dominance conditions with nonstrict inequalities, the singularity/nonsingularity problem is solved. Also the new sufficient diagonal dominance conditions are extended to the so-called mixed conditions, simultaneously involving the deleted row and column sums of an arbitrary finite set of matrices diagonally conjugated to a given one, which, in the simplest nontrivial case, reduce to the old-known Ostrowski conditions $$\left| {a_{ii} } \right| > \left( {\sum\limits_{j \ne i} {\left| {a_{ij} } \right|} } \right)^\alpha \left( {\sum\limits_{j \ne i} {\left| {a_{ji} } \right|} } \right)^{1 - \alpha } $$ , i = 1, ..., n, 0 ≤ α ≤ 1. The nonsingularity conditions obtained are used to provide new eigenvalue inclusion sets, depending on r, which, as r varies from 0 to n, serve as a bridge connecting the union of Gerschgorin’s disks with the Brualdi inclusion set. Bibliography: 16 titles. [ABSTRACT FROM AUTHOR]

The paper presents new upper and lower bounds for the singular values of rectangularmatrices explicitly involving the matrix sparsity pattern. These bounds are based on an upper bound for the Perron root of a nonnegative matrix and on the sparsity-dependent version of the Ostrowski-Brauer theorem on eigenvalue inclusion regions. Bibliography: 7 titles. [ABSTRACT FROM AUTHOR]

The paper presents new upper and lower bounds for the Perron root of a nonnegative matrix in terms of the simple circuits of length not exceeding k and the simple paths of length k, 1 ≤ k ≤ n, in the directed graph of the matrix. For each k, 1 ≤ k ≤ n, these bounds are intermediate between the circuit bounds and the path-dependent bounds suggested previously, and for k = 1 and k = n they reduce to the corresponding path-dependent bounds and the circuit bounds, respectively. Bibliography: 5 titles. [ABSTRACT FROM AUTHOR]

The paper presents new pseudoblock diagonal-dominance conditions for matrices with fixed block partitioning, which generalize the pointwise kth-order diagonal-dominance conditions and pointwise circuit diagonal-dominance conditions. For matrices satisfying pseudoblock conditions, the singularity/nonsingularity problem is considered. In particular, for block 2 × 2 matrices, certain known results are improved and generalized. Some eigenvalue inclusion regions corresponding to pseudoblock nonsingularity conditions are presented. Bibliography: 11 titles. [ABSTRACT FROM AUTHOR]

The paper considers the singularity/nonsingularity problem for matrices satisfying certain conditions of diagonal dominance. The conditions considered extend the classical diagonal dominance conditions and involve the directed graph of the matrix in question. Furthermore, in the case of the so-called mixed diagonal dominance, the corresponding conditions are allowed to involve both row and column sums for an arbitrary finite set of matrices diagonally conjugated to the original matrix. Conditions sufficient for the nonsingularity of quasi-irreducible matrices strictly diagonally dominant in certain senses are established, as well as necessary and sufficient conditions of singularity/nonsingularity for weakly diagonally dominant matrices in the irreducible case. The results obtained are used to describe inclusion regions for eigenvalues of arbitrary matrices. In particular, a direct extension of the Gerschgorin (r = 1) and Ostrowski-Brauer (r = 2) theorems to r ≥ 3 is presented. Bibliography: 18 titles. [ABSTRACT FROM AUTHOR]

The paper completely solves the problem of optimal diagonal scaling for quasireal Hermitian positive-definite matrices of order 3. In particular, in the most interesting irreducible case, it is demonstrated that for any matrix C from the class considered there is a uniquely determined optimally scaled matrix DCD0 of one of the four canonical types. Formulas for the entries of the diagonal matrix D0 are presented, as well as formulas for the eigenvalues and eigenvectors of DCD0 and for the optimal condition number of C, which is equal to k(DCD0). The optimality of the Jacobi scaling is analyzed. Bibliography: 10 titles. [ABSTRACT FROM AUTHOR]

Let an n × n Hermitian matrix A be presented in block 2 × 2 form as $$A = \left[ {\begin{array}{*{20}c} {A_{11} } & {A_{12} } \\ {A_{12}^* } & {A_{22} } \\ \end{array} } \right]$$ , where A12 ≠ 0, and assume that the diagonal blocks A11 and A22 are positive definite. Under these assumptions, it is proved that the extreme eigenvalues of A satisfy the bounds where $$R = A_{11}^{ - 1/2} A_{12} A_{22}^{ - 1/2}$$ and ‖ ⋅ ‖ is the spectral norm. Further, in the positive-definite case, several equivalent conditions necessary and sufficient for both of the above bounds to be attained are provided. Bibliography: 6 titles. [ABSTRACT FROM AUTHOR]

Lower and upper conditional bounds for the eigenvalues of a Hermitian positive-definite block 2 × 2 matrix, describing their closest possible location, are derived. The notions of partially and totally optimal spectra are introduced, and several equivalent characterizations of matrices with partially and totally optimal spectra are presented. It is shown that the block 2 × 2 block Jacobi scaled matrices and the so-called equilibrated matrices have totally optimal spectra. Copyright © 2005 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The paper suggests two-sided, upper, and lower circuit bounds for the Perron root of a nonnegative matrix, most of which are derived based on an extension of the monotonicity property of the Perron root established by Fiedler and Pták. Bibliography:9titles. [ABSTRACT FROM AUTHOR]

The lower boundfor the difference of the extreme eigenvalues of an n × n Hermitian block 2 × 2 matrix A =is established, and conditions necessary and sufficient for this bound to be attained at A are provided. Some corollaries of this result are derived. In particular, for a positive-definite matrix A, it is demonstrated that ?1 (A) - ?n (A) = 2?A12? if and only if A is optimally conditioned, and explicit expressions for the extreme eigenvalues of such matrices are obtained. Bibliography:5titles. [ABSTRACT FROM AUTHOR]