David Carlson

What are Schur complements, anyway?

Abstract Given a matrix M= A B C D , the Schur complements of A in M are the matrices of the form S = D − CaB, where a is a generalized inverse of A. We survey several recent characterizations of Schur complements, and discuss where they arose and how they are related.

The Linear Algebra Curriculum Study Group Recommendations for the First Course in Linear Algebra

There is a growing concern that the linear algebra curriculum at many schools does not adequately address the needs of the students it attempts to serve. In recent years, demand for linear algebra training has risen in client disciplines such as engineering, computer science, operations research, economics, and statis? tics. At the same time, hardware and software improvements in computer science have raised the power of linear algebra to solve problems that are orders of magnitude greater than dreamed possible a few decades ago. Yet in many courses, the importance of linear algebra in applied fields is not communicated to students, and the influence of the computer is not felt in the classroom, in the selection of topics covered or in the mode of presentation. Furthermore, an overemphasis on abstraction may overwhelm beginning students to the point where they leave the course with little understanding or mastery of the basic concepts they may need in later courses and their careers.

Teaching Linear Algebra: Must the Fog Always Roll In?.

(1993). Teaching Linear Algebra: Must the Fog Always Roll in? The College Mathematics Journal: Vol. 24, No. 1, pp. 29-40.

Generalized minimax and interlacing theorems

We give some steps towards a unified theory of Courant-Fischer minimax-type formulas and Cauchy interlacing-type inequalities that have been obtained for the eigenvalues of Hermitian matrices, for singular values of complex matrices, and for invariant factors of integral matrices We also unify and extend work on eigenvalues, singular values, and invariant factors of pairs of matrices and their sum or product

Block diagonal semistability factors and Lyapunov semistability of block triangular matrices

Abstract The concept of Lyapunov diagonal (semi)stability is generalized to the block diagonal case, unifying the theory of Lyapunov diagonal stability and Lyapunov stability. The corresponding generalizations of the concepts of maximal Lyapunov scaling factors are applied to study the Lyapunov semistability of block triangular matrices.

Nonsingularity criteria for matrices involving combinatorial considerations

Abstract We survey related nonsingularity criteria for matrices which involve combinatorial considerations, generally involving combinatorial restrictions on the matrices considered and conditions on various graphs of those matrices. These criteria include the existence of nonzero diagonal products for matrices over arbitrary fields; sign-nonsingularity (due to J. Maybee), potential nonsingularity, and stable nonsingularity (due to W. Anderson), all for real matrices; and some recent work for matrices over arbitrary fields involving a closure operator on matrices (due to the author and D. Hershkowitz).

Common eigenvectors and quasicommutativity of sets of simultaneously triangularizable matrices

Abstract A set of simultaneously triangularizable square matrices over an arbitrary field is considered. If the matrices are also quasicommutative, then they have a common eigenvector for every distinct set of corresponding eigenvalues. Conversely, if the set of matrices has this common eigenvector property hereditarily (i.e., for every set of corresponding blocks in every simultaneous block triangularization), then the matrices are quasicommutative.

The Teaching and Learning of Tertiary Algebra

This chapter reports on some current educational issues related to the teaching and learning of tertiary algebra—in particular, abstract algebra, discrete mathematics, linear algebra, and number theory. The causes of conceptual difficulties experienced by many students are identified and possible ways of overcoming them, sometimes using a specific pedagogical framework, are discussed. Issues related to students’ motivation are explored and pedagogical possibilities for overcoming some of the problems in both these areas are also explored. This report also addresses issues associated with the dissemination of educational work to tertiary instructors who are typically mathematicians rather than mathematics educators. Furthermore, the role of computers in tertiary algebra courses is considered, focusing on the use of Computer Algebra Systems (at the tertiary level) and the use of the programming language ISETL that helps students construct and work with algebraic objects. This chapter makes recommendations for improving practices for teaching tertiary algebra and proposes areas for further research.

Extremal patterns of distinct entries in vectors in the range of a matrix

For an m × n matrix A over a field F we consider the following quantities: μ(A), the maximum multiplicity of a field element as a component of a nonzero vector in the range of A, and δ(A), the minimum number of distinct entries in a nonzero vector in the range of A. In terms of ramk(A), we describe the set of possible values of μand δ and discuss the possible relations between them. We also develop a general affine geometric structure in which the sets of values of μ and δ may be characterized linear algebraically.

Nonsingularity Criteria for General Combinatorially Symmetric Matrices

We present two criteria for nonsingularity of matrices over general fields. The first applies to irreducible acyclic matrices with zero diagonal, the second to arbitrary combinatorially symmetric matrices. Both involve graphs of matrices and a closure operation on sets of vertices of the graphs. We focus attention not on the entries of the matrices under consideration but rather on patterns of zero entries of null vectors of the matrices. Our criteria are of theoretical rather than of computational interest.