LINEAR ALGEBRA SEMINAR
Tuesday 4:00-5:00 p.m. Parker 352
April 16, 2013
Speaker: T.Y. Tam (joint work with Peter Nylen, to be completed)
Title: Similarity of Inverse and Adjoint
Abstract: Given a nonsingular matrix A, we discuss when the inverse of A and the complex conjugate transpose of A are similar. The result is known (for example Horn and Johnson's Matrix Analysis). We would like to extend the result to semisimple Lie group via complete multiplicative Jordan Decomposition (CMJD).
April 9, 2013
Speaker: Frank Uhlig (joint work with Froilán Dopico, Madrid)
Title: The Matrix Symmetrizer Problem is an Eigenvalue Problem
Abstract: By a theorem of Frobenius (1910), every matrix A_n,n over any field F is the product of two symmetric ones, i.e., A = S1 · S2 where S1 can be chosen nonsingular. Here we detail the history of the symmetric matrix factorization and of factorizations in general. Then, using the algorithm of Huang and Nong (2010) for linear systems, we develop an iterative algorithm to compute a symmetric matrix S = S^T ∈ F_n,n for which S · A is symmetric for any given square matrix A ∈ F_n,n where F = R or C . Computing such a ’symmetrizer’ S from A has been a daunting task and was not possible before. In this method, the symmetrizer problem is viewed as a linear equations problem : Find a nonsingular symmetric S so that S · A = AT · S. However, unlike the Huang-Nong algorithm, this problem is best solved via eigenanalyses of A and the SVD that lets us find a Jordan normal form of A stably. Matlab codes support these claims and now make matrix symmetrizers readily available.
PDF available here
April 2, 2013
Dr. Huang will continue the last seminar's talk, to explore some results related to CMJD.
Speaker: Huajun Huang
Title: On complete multiplicative Jordan decomposition
Abstract: The matrix eigenvalue moduli has an analog in semisimple Lie groups, which is determined by the complete multiplicative Jordan decomposition (CMJD). A variety of properties and applications of CMJD will be presented in this talk.
March 25, 2013
Speaker: Huajun Huang
Title: On Complete multiplicative Jordan decomposition
Abstract: The matrix eigenvalue moduli has an analog in semisimple Lie groups, which is determined by the complete multiplicative Jordan decomposition (CMJD). A variety of properties and applications of CMJD will be presented in this talk.
March 5, 2013
NO SEMINAR TODAY
March 12. 2013
NO SEMINAR -- SPRING BREAK
February 26, 2013
Speaker: Dr. Peng Zeng
Title: Interface between Statistics and Linear Algebra
Abstract: The research in Statistics is closely related to linear algebra. In this talk, we will first review some classic methods in Statistics and comment on their connections to linear algebra. After that, we will discuss some recent developments in Statistics that are also interesting in linear algebra.
February 19, 2013
Speaker: Dr. Tufan Kuzpinari, a visiting scholar hosted by Erkan Nane
Title: Three-crossed modules
Abstract: We introduce the notion of 3-crossed modules, which extends the notions of 1-crossed module (Whitehead) and 2-crossed module (Conduché). We show that the category of 3-crossed modules is equivalent to the category of simplicial groups having a Moore complex of length 3. We make explicit the relationship with the cat3-groups (Loday) and the 3-hyper-complexes (Cegarra-Carrasco), which also model algebraically homotopy 4-types.
February 12, 2013
Speaker: Prof. Thomas Pate
Title: Holder Type Inequalities for Non-symmetric Non-square Matrices
Abstract: Click here
February 5, 2013 NO SEMINAR TODAY
January 29, 2013
Speaker: Dean Hoffman
Title: On the Centralizer of a Matrix.
Abstract: I'm a big fan of very short abstracts; in this case, the title itself tells it all!
January 22, 2013
Speaker: Doug Leonard
Title: The art of row reduction
Abstract: Since row reduction does not affect the row space of a matrix but destroys the column space, I have chosen to teach linear algebra from a row vector/row reduction perspective, which I find much more natural than a column vector/row reduction one.
1) I will give basic examples (at the level of MATH 2660) showing why I like this approach for bases, matrices for linear transformations, change of bases, kernels and images, and eigenspaces.
2) Then I will do modified Gram-Schmidt as row reduction, producing a non-standard integer QR decomposition for integer matrices.
3) I will solve systems of linear equations in terms of the generalization in commutative algebra to Gröbner bases and varieties.
4) And, time permitting, I will look at examples of Macaulay2 doing column reduction on generating sets of column vectors for modules over polynomial rings.
For full presentation PDF, click here
November 13, 2012
Speaker: Prof. Thomas H. Pate
Title: Immanants Orderings and Inequalities Involving Generalized Matrix Functions
Abstract: I will discuss the partial ordering induced upon the set of irreducible characters of the symmetric groups via the set of Hermitian positive semi-definite matrices.
For full presentation pdf, click here
November 6, 2012
Speaker: Avantha Indika
Title: Orthogonal bases of Brauer symmetry classes of tensors for the dihedral group.
Abstract: We will give the necessary and sufficient conditions for the existance of an orthogonal basis consisting of standard decomposable tensors of a symmetry class of tensors associated with a Brauer character of the dihedral group.
For full presentation pdf, click here
October 30, 2012
Speaker: Tin-Yau Tam
Title: Marcus-Oliveira Conjecture on the Determinants of the Orbit Sum
Abstract: We will review Fiedler's result on the determinant of sum of two Hermitian matrices. Then we discuss analogous results of Tin-Yau Tam and Mary Clair Thompson. Marcus-Oliveira conjecture will be introduced. If you Google by the key words: Marcus Oliveira conjecture, you will get a lot of information on the web.
For full presentation pdf, click here
October 23, 2012
Speaker: Peter Nylen
Title: Realizations of Interlacing by Structured Matrices
Abstract available here
October 16, 2012 NO SEMINAR
October 9, 2012
Speaker: Daniel Brice
Title: Derivations of Parabolic Subalgebras II
Abstract: Consider p, a parabolic subalgebra of a complex semisimple Lie algebra g. We will show that every derivation of p is of the form ad z for some z in p. If time permits, we will also prove the real case.
October 2, 2012
Speaker: Douglas Leonard
Title: Thinking outside the matrix
Abstract available here
September 25, 2012
Speaker: Daniel Brice
Title: Der(g) where g is parabolic
Abstract: Let g be a parabolic subalgebra of a finite-dimensional reductive Lie algebra L over the real or complex field. The derivation algebra Der(g) consists of linear endomorphisms f of g that satisfy f([x,y])=[f(x),y]+[x,f(y)] for all x, y in g. The speakers examine the structure of Der(g), especially as it relates to the grading on g given by the root space decomposition of L. The speakers will give a short survey of existing related literature as well as present some new results. The presentation will be accessible to a general audience familiar with basic linear algebra techniques.
For full presentation PDF, click here
September 18, 2012
Speaker: Dr. Wen Yan (Tuskegee University)
Title: Anti-Diagonals of Symmetric and Skew-symmetric Matrices with Prescribed Eigenvalues
Abstract: The complete relationship between the k-th anti-diagonals and the eigenvalues of an n-by-n real symmetric matrix is obtained. When k is even, the relationship is weak majorization. The Hermitian case is also studied. Similar results are obtained for real and complex skew symmetric matrices.
For full presentation PDF, click here
September 11, 2012
Speaker: Wen Yan (Tuskegee University)
Title: Anti-Diagonals of Symmetric and Skew-symmetric Matrices with Prescribed Eigenvalues
Abstract: The complete relationship between the k-th anti-diagonals and the eigenvalues of an n-by-n real symmetric matrix is obtained. When k is even, the relationship is weak majorization. The Hermitian case is also studied. Similar results are obtained for real and complex skew symmetric matrices.
September 4, 2012
Speaker: Mary Clair Thompson
Title: The Gelfand-Naimark Decomposition
Abstract: We decompose the Lie group G with the Gelfand-Naimark Decomposition, and iterate the process to create a sequence. We investigate the convergence of the sequence for certain elements of G.
August 28, 2012
Speaker: Mary Clair Thompson (advisor: Tin-Yau Tam).
Title: The Convergence of the Iterated Aluthge Transform