A note on unsolvable systems of max–min (fuzzy) equations

For unsolvable systems of linear equations of the form Ax=b over the max–min (fuzzy) algebra

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we propose an efficient method for finding a Chebychev-best approximation of the matrix in the set

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.     

General preservers of invariant subspace lattices

Let be the space of all bounded linear operators on a Banach space X and let LatA be the lattice of invariant subspaces of the operator . We characterize some maps with one of the following preserving properties: Lat(Φ(A)+Φ(B))=Lat(A+B), or Lat(Φ(A)Φ(B))=Lat(AB), or Lat(Φ(A)Φ(B)+Φ(B)Φ(A))=Lat(AB+BA), or Lat(Φ(A)Φ(B)Φ(A))=Lat(ABA), or Lat([Φ(A),Φ(B)])=Lat([A,B]).     

Characterizations of some classes of relatively regular operators

A bounded linear operator A on a Banach space is called relatively regular, if there is a bounded linear operator B such that ABA=A. In this case B is called a g₁-inverse of A. In this paper we characterize some classes of relatively regular operators A via the set {B₁-B₂:B₁ and B₂ are g₁-inverses of A}.     

Nonlinear maps preserving Lie products on factor von Neumann algebras

In this paper, we prove that every bijective map preserving Lie products from a factor von Neumann algebra into another factor von Neumann algebra is of the form A→ψ(A)+ξ(A), where is an additive isomorphism or the negative of an additive anti-isomorphism and is a map with ξ(AB-BA)=0 for all .     

An atlas for tridiagonal isospectral manifolds

Let be the compact manifold of real symmetric tridiagonal matrices conjugate to a given diagonal matrix Λ with simple spectrum. We introduce bidiagonal coordinates, charts defined on open dense domains forming an explicit atlas for . In contrast to the standard inverse variables, consisting of eigenvalues and norming constants, every matrix in now lies in the interior of some chart domain. We provide examples of the convenience of these new coordinates for the study of asymptotics of isospectral dynamics, both for continuous and discrete time.     

A note on an unusual type of polar decomposition

Motivated by applications in the theory of unitary congruence, we introduce the factorization of a square complex matrix A of the form A=SU, where S is complex symmetric and U is unitary. We call this factorization a symmetric–unitary polar decomposition or an SUPD. It is shown that an SUPD exists for every matrix A and is always nonunique. Even the symmetric factor S can be chosen in infinitely many ways. Nevertheless, we show that many properties of the conventional polar decomposition related to normal matrices have their counterparts for the SUPD, provided that normal matrices are replaced with conjugate–normal ones.     

Diameter preserving surjections in the geometry of matrices

We consider a class of graphs subject to certain restrictions, including the finiteness of diameters. Any surjective mapping φ:Γ→Γ^′ between graphs from this class is shown to be an isomorphism provided that the following holds: Any two points of Γ are at a distance equal to the diameter of Γ if, and only if, their images are at a distance equal to the diameter of Γ^′. This result is then applied to the graphs arising from the adjacency relations of spaces of rectangular matrices, spaces of Hermitian matrices, and Grassmann spaces (projective spaces of rectangular matrices).     

Equality of higher numerical ranges of matrices and a conjecture of Kippenhahn on Hermitian pencils

Let M_n be the algebra of all n × n complex matrices. For 1 k n, the kth numerical range of A M_n is defined by W_k(A) = (1/k)∑_j^k=1x_j^*Ax_j : x₁, …, x_k is an orthonormal set in ⁿ]. It is known that tr A/n = W_n(A) W_n−1(A) W₁(A). We study the condition on A under which W_m(A) = W_k(A) for some given 1 m < k n. It turns out that this study is closely related to a conjecture of Kippenhahn on Hermitian pencils. A new class of counterexamples to the conjecture is constructed, based on the theory of the numerical range.     

A formula for the number of Latin squares

We establish an explicit formula for the number of Latin squares of order n:

, where B_n is the set of n×n(0,1) matrices, σ₀(A is the number of zero elements of the matrix A and per A is the permanent of the matrix A.     

Linear operators preserving unitarily invariant norms of matrices

Let F_{m × n} be the set of all m × n matrices over the field F = C or R Denote by U_n(F) the group of all n × n unitary or orthogonal matrices according as F = C or F-R. A norm N() on Fm ×n, is unitarily invariant if N(UAV) = N(A): for all A ∈ F _m×n U ∈ U_m(F). and V ∈ U_n(F). We characterize those linear operators TF_{m × n} → F_{m × n}which satisfy N (T(A)) = N(A)for all A ∈ F_{m × n}

for a given unitarily invariant norm N(). It is shown that the problem is equivalent to characterizing those operators which preserve certain subsets in F_{m × n} To develop the theory we prove some results concerning unitary operators on F_{m × n} which are of independent interest.     

Finding all essential terms of a characteristic maxpolynomial

Let us denote ab=max(a,b) and ab=a+b for and extend this pair of operations to matrices and vectors in the same way as in linear algebra. We present an O(n²(m+n log n)) algorithm for finding all essential terms of the max-algebraic characteristic polynomial of an n×n matrix over with m finite elements. In the cases when all terms are essential, this algorithm also solves the following problem: Given an n×n matrix A and k{1,…,n}, find a k×k principal submatrix of A whose assignment problem value is maximum.     

Hypernormal matrices

An nxn matrix A is hypernormal if APA^*=A^*PA for all permutation matrices P. We shall explain how to construct hypernormal matrices.     

Sign balance for finite groups of Lie type

A product formula for the parity generating function of the number of 1’s in invertible matrices over is given. The computation is based on algebraic tools such as the Bruhat decomposition. It is somewhat surprising that the number of such matrices with odd number of 1’s is greater than the number of those with even number of 1’s. The same technique can be used to obtain a parity generating function also for symplectic matrices over . We present also a generating function for the sum of entries of matrices over an arbitrary finite field calculated in . The Mahonian distribution appears in these formulas.     

The construction of irreducible normalized representations of the general linear group

A method is described for constructing in an explicit form an irreducible representation T of M_n(F), the set of all n × n matrices over the real or complex field F, satisfying the condition T(A^*)=T^*(A) for all A∈M_n(F).     

Atoms of set systems with a fixed number of pairwise unions

For a finite set system with ground set X, we let

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. An atom of H is a nonempty maximal subset C of X such that for all A H, either C A or C ∩ A = 0. We obtain a best possible upper bound for the number of atoms determined by a set system H with H = k and H H = u for all integers k and u. This answers a problem posed by Sós.     

The rank of the difference of matrices with prescribed similarity classes

Let F be a field and let A,B be n × n matrices over I. We study the rank of A' - B' when A and B run over the set of matrices similar to A and B, respectively.     

The growth of laguerre matrix polynomials on bounded intervals

Let A be a matrix in ^r×r such that Re(z) > −1/2 for all the eigenvalues of A and let {π_n^(A,1/2) (x)} be the normalized sequence of Laguerre matrix polynomials associated with A. In this paper, it is proved that π_n^(A,1/2) (x) = O(n^(A)/2ln^r−1(n)) and π_n+1^(A,1/2) (x) − π_n^(A,1/2) (x) = O(n^((A)−1)/2ln^r−1(n)) uniformly on bounded intervals, where (A) = max{Re(z); z eigenvalue of A}.     

A result about determinantal divisors

Let Rbe a principal ideal ringR_n the ring of n× nmatrices over R, and d_k(A) the kth determinantal divisor of Afor 1 ≤ k≤ n, where Ais any element of R_n, It is shown that if A,BεR_n, det(A) det(B:) ≠ 0, then d_k(AB) ≡ 0 mod d_k(A) d_k(B). If in addition (det(A), det(B)) = 1, then it is also shown that d_k(AB) = d_k(A) d_k(B). This provides a new proof of the multiplicativity of the Smith normal form for matrices with relatively prime determinants.     

Solution of bordered singular systems in numerical continuation and bifurcation

In numerical continuation and bifurcation problems linear systems with coefficient matrices in the block form arise naturally. Here

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and n may be large but m is small. A usually has a special structure (banded, block banded, sparse,…) and B, C, D are dense, so that it is advisable to use a specialized solver for A and to solve with M by some block method. Unfortunately, A is often also a nearly singular matrix (in fact, made nonsingular only by roundoff and truncation errors). On the other hand, M is usually nonsingular but can be ill-conditioned and in certain situations will degenerate to singularity as well. We describe numerical tests for this problem using the mixed block elimination method of Govaerts and Pryce (1993) for solving bordered linear systems with possibly nearly singular blocks A. To this end, we compute by Newton's method a triple-point bifurcation point in a parameterized reaction—diffusion equation (the Brusselator). The numerical tests show that the linear systems are solved in a stable way, in spite of the use of a black-box solver (SGBTRS from LAPACK) for a nearly singular matrix.     

On k-spaces of real matrices

It is shown that if W is a linear subspace of real n × n matrices, such that rank (A) = k for all 0 ≠ A ∈ W, then dim W≤ n. If dim W = n.5≤ n is prime, and 2 is primitive modulo n then k =1.