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# Linear Algebra

DMS Linear Algebra/Algebra Seminar
Apr 17, 2018 04:00 PM
Parker Hall 352

Speaker: Tin-Yau Tam

Title: Matrix Inequalities and Their Extensions to Lie Groups

Abstract: We will discuss some classical matrix inequalities and their extensions that are topics in my new book Matrix Inequalities and Their Extensions to Lie Groups.

DMS Linear Algebra/Algebra Seminar
Apr 03, 2018 04:00 PM
Parker Hall 352

Speaker: Avantha Indika

Title: A study of the structure of the symmetry classes

Abstract: Will discuss the studies on standard (decomposable) symmetrized tensors to understand the structure of symmetry classes of tensors associated with finite groups and the use of coset space to understand the geometric structure of a symmetry class of tensors.

DMS Linear Algebra/Algebra Seminar
Mar 20, 2018 04:00 PM

Speaker: Zejun Huang (Hunan University, China)

Title: Preserver problems on matrices

Abstract: Preserver problems on matrices concern the characterization of linear or nonlinear maps on matrices that leave certain properties invariant. In this talk, I will present some results on both linear and nonlinear preserver problems on matrices.

DMS Linear Algebra/Algebra Seminar
Mar 06, 2018 04:00 PM
Parker Hall 352

Speaker: Jason Liu

Title: Toeplitz Matrices, Symmetric Matrices, and Unitary Similarity

Abstract: In this talk, I will present the result that every Toeplitz matrix is uniformly unitarily similar to some complex symmetric matrices via a unitary matrix. Conversely, every symmetric matrix is unitarily similar to some Toeplitz matrix when $$n < 4$$.   I will relate this result with the facts of every matrix is a product of some Toeplitz matrices, which was proved by Ye and Lim in 2015, and a super-fast divide-and-conquer method to make the eigenproblem of Toeplitz matrices with nearly linear complexity presented by Vogel et al. in 2016. I will also present that every multilevel Toeplitz matrix is unitarily similar to a symmetric matrix.

DMS Linear Algebra/Algebra Seminar
Feb 27, 2018 04:00 PM
Parker Hall 352

Speaker: Zhuoheng He

Title: The general $$\phi$$-Hermitian solution to mixed pairs of quaternion matrix Sylvester equations

Abstract: In this talk, we consider two systems of mixed pairs of quaternion matrix Sylvester equations

$A_{1}X-YB_{1}=C_{1},~A_{2}Z-YB_{2}=C_{2}$ and $A_{1}X-YB_{1}=C_{1},~A_{2}Y-ZB_{2}=C_{2},$

where $$Z$$ is $$\phi$$-Hermitian. Some practical necessary and sufficient conditions for the existence of a solution $$(X,Y,Z)$$ to those systems in terms of the ranks and Moore-Penrose inverses of the given coefficient matrices will be presented. Moreover, the general solutions to these systems are explicitly given when they are solvable. We also provide some numerical examples to illustrate our results.

DMS Linear Algebra/Algebra Seminar
Feb 20, 2018 04:00 PM
Parker Hall 352

Speaker: Frank Uhlig

Title: The Eight Epochs of Math as regards past and future Matrix Computations

Abstract:  This survey paper gives a personal assessment of Epoch making advances in Matrix Computations, from antiquity and with an eye towards tomorrow.  It traces the development of number systems and elementary algebra and the uses of Gaussian Elimination methods from around 4000 BC on to current real-time Neural Network computations to solve time-varying matrix equations. The paper includes relevant advances from China from the 3rd century AD on and from India and Persia in the 9th and later centuries. Then it discusses the conceptual genesis of vectors and matrices in central Europe and in Japan in the 14th through 17th centuries AD. Followed by the 150 year cul-de-sac of polynomial root finder research for matrix eigenvalues, as well as the superbly useful matrix iterative methods and Francis’s matrix eigenvalue algorithm from the last century. Finally we explain the recent use of initial value problem solvers and high order 1-step ahead discretization formulas to master time-varying linear and nonlinear matrix equations via Zhang Neural Networks. This paper ends with a short outlook upon new hardware schemes with multilevel processors that go beyond the 0-1 base 2 framework which all of our past and current electronic computers have been using.

DMS Linear Algebra/Algebra Seminar
Feb 13, 2018 04:00 PM
Parker Hall 352

Speaker: Luke Oeding

Title: Higher Order Partial Least Squares and an Application to Neuroscience.

Abstract: Partial least squares (PLS) is a method to discover a functional dependence between two sets of variables X and Y. PLS attempts to maximize the covariance between X and Y by projecting both onto new subspaces. Higher order partial least squares (HOPLS) comes into play when the sets of variables have additional tensorial structure. Simultaneous optimization of subspace projections may be obtained by a multilinear singular value decomposition (MSVD). I'll review PLS and SVD, and explain their higher order counterparts. Finally I'll describe recent work with G. Deshpande, A. Cichocki, D. Rangaprakash, and X.P. Hu where we propose to use HOPLS and Tensor Decompositions to discover latent linkages between EEG and fMRI signals from the brain, and ultimately use this to drive Brain Computer Interfaces (BCI)'s with the low size, weight and power of EEG, but with the accuracy of fMRI.

DMS Linear Algebra/Algebra Seminar
Feb 06, 2018 04:00 PM
Parker Hall 352

Speaker: Samir Raouafi

Title: A survey of Crouzeix's Conjecture

DMS Linear Algebra/Algebra Seminar
Jan 30, 2018 04:00 PM
Parker Hall 352

Speaker: Mehmet Gumus

Title: On the Lyapunov-type diagonal stability

Abstract: In this talk, we present several recent developments regarding Lyapunov diagonal stability. This type of matrix stability plays an important role in various applied areas such as population dynamics, systems theory, complex networks, and mathematical economics. First, we examine a result of Redheer that reduces Lyapunov diagonal stability of a matrix to common diagonal Lyapunov solutions on two matrices of order one less. An enhanced statement of this result based on the Schur complement formulation is presented here along with a shorter and purely matrix-theoretic proof. We develop a number of extensions to this result and formulate the range of feasible common diagonal Lyapunov solutions. In particular, we derive explicit algebraic conditions for a set of 2x2 matrices to share a common diagonal Lyapunov solution. Second, we present a new characterization involving Hadamard multiplications for simultaneous Lyapunov diagonal stability on a set of matrices. This extends a useful characterization, due to Kraaijevanger, of Lyapunov diagonal stability in terms of the P-matrix property under similar Hadamard multiplications. Our development mainly relies on a new notion called P-sets, which is a generalization of P-matrices, and a recent result of Berman, Goldberg, and Shorten.

DMS Linear Algebra/Algebra Seminar
Jan 23, 2018 04:00 PM
Parker Hall 352

Speaker: Huajun Huang

Title: A survey of the Marcus-de Oliveira determinantal conjecture

Abstract: The Marcus-de Oliveira determinantal conjecture (MOC) says that the complex number det(A+B), where A and B are normal matrices, is located in the convex hull of the determinants of sum of diagonal matrices similar to A and B, respectively.  I will survey in this talk the history, progress, and challenges of this conjecture, and introduce the related geometric properties and extended results to the determinants of sums of matrices.

Last Updated: 08/21/2017