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For a block upper triangular matrix, a necessary and sufficient condition has been given to let it be the sum of block upper rectangular matrices satisfying certain rank constraints; see [12] . The proof involves elements from Integer Programming (totally unimodular systems of equations playing a role in particular) and employs Farkas' Lemma. The linear space of block upper triangular matrices can be viewed as being determined by a special pattern of zeros. The present paper is concerned with the question whether the decomposition result can be extended to situations where other, less restrictive, zero patterns play a role. It is shown that such generalizations do indeed hold for certain directed graphs determining the pattern of zeros. The graphs in question are what will be called L -free. This notion is akin to other graph theoretical concepts available in the literature, among them the one of being N -free in the sense of [16] .

A zero pattern algebra is a matrix algebra determined by a pattern of zeros corresponding to a preorder (i.e., a reflexive transitive digraph). The issue studied in [6] is: under what conditions is a matrix in a zero pattern algebra A a sum of (rank one) idempotents in A or a logarithmic residue in A ? Here logarithmic residues are contour integrals of logarithmic derivatives of analytic A -valued functions. As has been established in [6] , there is a necessary condition involving certain rank/trace requirements. Algebras for which this necessary condition is also sufficient are said to be rank/trace complete. Several classes of rank/trace complete algebras are identified in [6] . Also operations are described there for producing rank/trace complete algebras out of given ones. A basic operation of that type is taking a disjoint union. In the present article this simple operation is generalized to a more involved one which preserves rank/trace completeness too. The new operation turns out to be a useful tool for establishing rank/trace completeness in situations that up to now could not be handled. One of the positive results is concerned with bouquets, a special type of rooted trees. It provides a partial answer to an issue raised in [7] .

A zero pattern algebra is a matrix subalgebra of C n × n determined by a pattern of zeros. The issue in this paper is: under what conditions is a matrix in a zero pattern algebra A a sum of (rank one) idempotents in A or a logarithmic residue in A ? Here logarithmic residues are contour integrals of logarithmic derivatives of analytic A -valued functions. It turns out that there is a necessary condition involving certain rank/trace requirements. Although these requirements are generally not sufficient, there are several important cases where they are.

Let B be a unital Banach algebra, which can in a certain sense be approximated by finite dimensional algebras. For instance, AF C ⁎ -algebras belong to this class. Further, let f be an analytic function on some bounded Cauchy domain Δ with values in B and suppose that the contour integral of the logarithmic derivative f ′ ( λ ) f − 1 ( λ ) along the positively oriented boundary ∂Δ vanishes (or is even only quasinilpotent). We prove that then f takes invertible values on all of Δ. This means that such Banach algebras are spectrally regular.

It is described how my contacts with a group of functional analysts in Chemnitz (Germany) came along and resulted in a longstanding cooperation on logarithmic residues and spectral regularity in a Banach algebra setting.

This obituary for Israel Gohberg consists of a general introduction, separate contributions of the six authors, all of whom worked closely with him, and a final note. The material gives an impression of the life of this great mathematician, of his monumental impact in the areas he worked in, of how he cooperated with colleagues, and of his ability to stimulate people in their mathematical activities.

A new class of Banach algebra valued functions is identified for which the logarithmic residue with respect to a Cauchy domain Δ vanishes (if and) only if the functions take invertible values in Δ . Trace conditions and the extraction of elementary factors of the type e - p + ( λ - α ) p play an important role. The class contains the Fredholm operator valued functions and the Banach algebra valued functions possessing a simply meromorphic resolvent as special instances. An example is given to show that new ground is covered and a long standing open problem is discussed from a fresh angle.

A necessary and sufficient condition is given for a block upper triangular matrix A to be the sum of block upper rectangular matrices satisfying certain rank constraints. The condition is formulated in terms of the ranks of certain submatrices of A. The proof goes by reduction to an integer programming problem. This integer programming problem has a totally unimodular constraint matrix which makes it possible to utilize Farkas' Lemma.

A connection is made between the Two Machine Flow Shop Problem (2MFSP) from job scheduling theory and the issue of quasicomplete factorization of rational matrix functions. A quasicomplete factorization is a factorization into elementary (i.e., degree one) factors such that the number of factors is minimal. For a companion based matrix function W, the number of factors in a quasicomplete factorization of W is related in a simple way to the minimum makespan of an instance J of 2MFSP which can be associated to W. As a consequence of this result, variants of the 2MFSP and other types of factorization can be related too.

A connection is made between two sets of problems. The first set involves factorization problems of specific rational matrix functions, the companion based matrix functions. The second set is concerned with variants of the two machine flow shop problem (2MFSP) from job scheduling theory. In particular, it is shown that with each companion based matrix function one can associate an instance of 2MFSP and vice versa. The latter can be done in such a way that the factorization properties of the companion based matrix function correspond to the combinatorial properties of the instance of 2MFSP.

Companion based matrix functions are rational matrix functions admitting a minimal realization involving state space matrices that are first companions. Necessary and sufficient conditions are given for a rational matrix function to be companion based. Minimal factorization of such functions is discussed in detail. It is shown that the property of being companion based is hereditary with respect to minimal factorization. Also, the issue of minimal factorization is reduced to a division problem for pairs of monic polynomials of the same degree. In this context, a connection with the Euclidean algorithm is made. The results apply to canonical Wiener-Hopf factorization as well as to complete factorization. The analysis of the latter leads to a combinatorial problem involving the eigenvalues of the state space matrices. The algorithmic aspects of this problem are intimately related to the two machine flow shop problem and Johnson's rule from job scheduling theory.

Let A be an arbitary (square) matrix. As is well known, there exists an invertible matrix S such that S-1AS is upper triangular. The present paper is concerned with the observation that S can always be chosen in the form S=∏L, where ∏ is a permutation matrix and L is lower triangular. Assuming that the eigenvalues of A are given, the matrices ∏, L, and U=L-1∏-1A∏L are constructed in an explicit way. The construction gives insight into the freedom one has in choosing the permutation matrix ∏. Two cases where ∏ can be chosen to be the identity matrix are discussed (A diagonable, A a low order Toeplitz matrix). There is a connection with systems theory.

For square matrices, the relationship is discussed between the notion of Schur complement and the (equivalent) concepts of matricial coupling and equivalence after extension. Special attention is paid to the Hermitian and skew-Hermitian cases.

A logarithmic residue is a contour integral of a logarithmic derivative (left or right) of an analytic Banach algebra valued function. For functions possessing a meromorphic inverse with simple poles only, the logarithmic residues are identified as the sums of idempotents. With the help of this observation, the issue of left versus right logarithmic residues is investigated, both for connected and nonconnected underlying Cauchy domains. Examples are given to elucidate the subject matter.

This paper is concerned with pairs of m × m matrices A.Z for which there exists an invertible m × m matrix S such that S−1 AS is upper triangular and S1ZS is lower triangular. Such pairs are said to admit simultaneous reduction to complementary triangular forms. In particular, if such a similariy S does not exist for the pair A.Z, tollowing question is taken into consideration: Does there exist a positive integer r, such that the pair of (m + r) × (m + r) matrices A O O O r Z O O O r admits simultaneous reduction to complementary triangular forms, so that the pair A.Z obtains the property after extension with zeroes? It is shown that the answer to this question is mixed. An example of a pair is given which obtains the property after extension with one zero. This example is put into contrast with a large number of results that answers the question negatively, i.e., that describes situations where the pair of extended matrices admits simultaneous reduction to complementary triangular forms if and only if the original pair does.

This paper is concerned with simultaneous reduction to triangular and companion forms of pairs of matrices A , Z with rank( I − AZ )=1. Special attention is paid to the case where A is a first and Z is a third companion matrix. Two types of simultaneous triangularization problems are considered: (1) the matrix A is to be transformed to upper triangular and Z to lower triangular form, (2) both A and Z are to be transformed to the same (upper) triangular form. The results on companions are made coordinate free by characterizing the pairs A , Z for which there exists an invertible matrix S such that S −1 AS is of first and S −1 ZS is of third companion type. One of the main theorems reads as follows: If rank( I − AZ )=1 and αζ ≠1 for every eigenvalue α of A and every eigenvalue ζ of Z , then A and Z admit simultaneous reduction to complementary triangular forms.

This paper is concerned with the following questions. Given a square matrix A , when does there exist an invertible lower triangular matrix L such that L -1 AL is upper triangular? And if so, what can be said about the order in which the eigenvalues of A may appear on the diagonal of L -1 AL ? The motivation for considering these questions comes from systems theory. In fact they arise in the study of complete factorizations of rational matrix functions. There is also an intimate connection with the problem of complementary triangularization of pairs of matrices discussed elsewhere by the first author.

The problem considered is that of simultaneous reduction to simple forms of pairs of upper triangular Toeplitz matrices. In this context, simple matrices are those that are equal to, or differ only slightly from, a power of the upper triangular nilpotent Jordan block. The problem is related to that of complementary triangularization of pairs of matrices and (therefore) has a background in systems theory. The main results are concerned with existence and uniqueness of the reduced form. Also a similarity invariant for pairs of upper triangular Toeplitz matrices is obtained. The invariant consists of a pair involving a complex number and an integer. The results can be generalized to a class of matrices strictly larger than that formed by the upper triangular Toeplitz matrices.