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# Linear Algebra

**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

Abstract: Please click here

**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.

**DMS Linear Algebra/Algebra Seminar**

Jan 16, 2018 04:00 PM

Parker Hall 352

Organizational Meeting

**DMS Linear Algebra/Algebra Seminar**

Dec 05, 2017 04:00 PM

Parker Hall 356

Speaker: **Sima Ahsani**

Title: Metrics on the manifold of positive definite matrices and matrix means

Abstract: I will talk about the set of positive definite matrices as a Riemannian manifold. Then I will introduce two metrics on this set and explain their basic properties. After describing their connection through the geometric mean of two positive definite matrices, similarities and differences between two metrics will be discussed.

**DMS Linear Algebra/Algebra Seminar**

Nov 28, 2017 04:00 PM

Parker Hall 356

Speaker: **Samir Raouafi**

Title: On the extension of the Kreiss Matrix Theorem

Abstract: Let A be a matrix with spectrum δ(A). The Kreiss Matrix Theorem (KMT), a well-known fact in applied matrix analysis, gives estimates of upper bounds for ∥A^{n}∥ if δ(A) is in the unit disc, or for ∥e^{tA}∥ if δ(A) is in the left-half plane based on the resolvent norm. In this talk, we shall discuss some extensions of this celebrated theorem to simply connected compact sets in the complex plane.

Last Updated: 08/21/2017