The Analytic Theory of Matrix Orthogonal Polynomials
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On Multivariate Interpolation
On Multivariate Interpolation Peter J. Olver† School of Mathematics University of Minnesota Minneapolis, MN 55455 U.S.A. [email protected] http://www.math.umn.edu/∼olver Abstract. A new approach to interpolation theory for functions of several variables is proposed. We develop a multivariate divided difference calculus based on the theory of non-commutative quasi-determinants. In addition, intriguing explicit formulae that connect the classical finite difference interpolation coefficients for univariate curves with multivariate interpolation coefficients for higher dimensional submanifolds are established. † Supported in part by NSF Grant DMS 11–08894. April 6, 2016 1 1. Introduction. Interpolation theory for functions of a single variable has a long and distinguished his- tory, dating back to Newton’s fundamental interpolation formula and the classical calculus of finite differences, [7, 47, 58, 64]. Standard numerical approximations to derivatives and many numerical integration methods for differential equations are based on the finite dif- ference calculus. However, historically, no comparable calculus was developed for functions of more than one variable. If one looks up multivariate interpolation in the classical books, one is essentially restricted to rectangular, or, slightly more generally, separable grids, over which the formulae are a simple adaptation of the univariate divided difference calculus. See [19] for historical details. Starting with G. Birkhoff, [2] (who was, coincidentally, my thesis advisor), recent years have seen a renewed level of interest in multivariate interpolation among both pure and applied researchers; see [18] for a fairly recent survey containing an extensive bibli- ography. De Boor and Ron, [8, 12, 13], and Sauer and Xu, [61, 10, 65], have systemati- cally studied the polynomial case. -
Orthogonal Reduction 1 the Row Echelon Form -.: Mathematical
MATH 5330: Computational Methods of Linear Algebra Lecture 9: Orthogonal Reduction Xianyi Zeng Department of Mathematical Sciences, UTEP 1 The Row Echelon Form Our target is to solve the normal equation: AtAx = Atb ; (1.1) m×n where A 2 R is arbitrary; we have shown previously that this is equivalent to the least squares problem: min jjAx−bjj : (1.2) x2Rn t n×n As A A2R is symmetric positive semi-definite, we can try to compute the Cholesky decom- t t n×n position such that A A = L L for some lower-triangular matrix L 2 R . One problem with this approach is that we're not fully exploring our information, particularly in Cholesky decomposition we treat AtA as a single entity in ignorance of the information about A itself. t m×m In particular, the structure A A motivates us to study a factorization A=QE, where Q2R m×n is orthogonal and E 2 R is to be determined. Then we may transform the normal equation to: EtEx = EtQtb ; (1.3) t m×m where the identity Q Q = Im (the identity matrix in R ) is used. This normal equation is equivalent to the least squares problem with E: t min Ex−Q b : (1.4) x2Rn Because orthogonal transformation preserves the L2-norm, (1.2) and (1.4) are equivalent to each n other. Indeed, for any x 2 R : jjAx−bjj2 = (b−Ax)t(b−Ax) = (b−QEx)t(b−QEx) = [Q(Qtb−Ex)]t[Q(Qtb−Ex)] t t t t t t t t 2 = (Q b−Ex) Q Q(Q b−Ex) = (Q b−Ex) (Q b−Ex) = Ex−Q b : Hence the target is to find an E such that (1.3) is easier to solve. -
On the Eigenvalues of Euclidean Distance Matrices
“main” — 2008/10/13 — 23:12 — page 237 — #1 Volume 27, N. 3, pp. 237–250, 2008 Copyright © 2008 SBMAC ISSN 0101-8205 www.scielo.br/cam On the eigenvalues of Euclidean distance matrices A.Y. ALFAKIH∗ Department of Mathematics and Statistics University of Windsor, Windsor, Ontario N9B 3P4, Canada E-mail: [email protected] Abstract. In this paper, the notion of equitable partitions (EP) is used to study the eigenvalues of Euclidean distance matrices (EDMs). In particular, EP is used to obtain the characteristic poly- nomials of regular EDMs and non-spherical centrally symmetric EDMs. The paper also presents methods for constructing cospectral EDMs and EDMs with exactly three distinct eigenvalues. Mathematical subject classification: 51K05, 15A18, 05C50. Key words: Euclidean distance matrices, eigenvalues, equitable partitions, characteristic poly- nomial. 1 Introduction ( ) An n ×n nonzero matrix D = di j is called a Euclidean distance matrix (EDM) 1, 2,..., n r if there exist points p p p in some Euclidean space < such that i j 2 , ,..., , di j = ||p − p || for all i j = 1 n where || || denotes the Euclidean norm. i , ,..., Let p , i ∈ N = {1 2 n}, be the set of points that generate an EDM π π ( , ,..., ) D. An m-partition of D is an ordered sequence = N1 N2 Nm of ,..., nonempty disjoint subsets of N whose union is N. The subsets N1 Nm are called the cells of the partition. The n-partition of D where each cell consists #760/08. Received: 07/IV/08. Accepted: 17/VI/08. ∗Research supported by the Natural Sciences and Engineering Research Council of Canada and MITACS. -
Determinants of Commuting-Block Matrices by Istvan Kovacs, Daniel S
Determinants of Commuting-Block Matrices by Istvan Kovacs, Daniel S. Silver*, and Susan G. Williams* Let R beacommutative ring, and Matn(R) the ring of n × n matrices over R.We (i,j) can regard a k × k matrix M =(A ) over Matn(R)asablock matrix,amatrix that has been partitioned into k2 submatrices (blocks)overR, each of size n × n. When M is regarded in this way, we denote its determinant by |M|.Wewill use the symbol D(M) for the determinant of M viewed as a k × k matrix over Matn(R). It is important to realize that D(M)isann × n matrix. Theorem 1. Let R be acommutative ring. Assume that M is a k × k block matrix of (i,j) blocks A ∈ Matn(R) that commute pairwise. Then | | | | (1,π(1)) (2,π(2)) ··· (k,π(k)) (1) M = D(M) = (sgn π)A A A . π∈Sk Here Sk is the symmetric group on k symbols; the summation is the usual one that appears in the definition of determinant. Theorem 1 is well known in the case k =2;the proof is often left as an exercise in linear algebra texts (see [4, page 164], for example). The general result is implicit in [3], but it is not widely known. We present a short, elementary proof using mathematical induction on k.Wesketch a second proof when the ring R has no zero divisors, a proof that is based on [3] and avoids induction by using the fact that commuting matrices over an algebraically closed field can be simultaneously triangularized. -
4.1 RANK of a MATRIX Rank List Given Matrix M, the Following Are Equal
page 1 of Section 4.1 CHAPTER 4 MATRICES CONTINUED SECTION 4.1 RANK OF A MATRIX rank list Given matrix M, the following are equal: (1) maximal number of ind cols (i.e., dim of the col space of M) (2) maximal number of ind rows (i.e., dim of the row space of M) (3) number of cols with pivots in the echelon form of M (4) number of nonzero rows in the echelon form of M You know that (1) = (3) and (2) = (4) from Section 3.1. To see that (3) = (4) just stare at some echelon forms. This one number (that all four things equal) is called the rank of M. As a special case, a zero matrix is said to have rank 0. how row ops affect rank Row ops don't change the rank because they don't change the max number of ind cols or rows. example 1 12-10 24-20 has rank 1 (maximally one ind col, by inspection) 48-40 000 []000 has rank 0 example 2 2540 0001 LetM= -2 1 -1 0 21170 To find the rank of M, use row ops R3=R1+R3 R4=-R1+R4 R2ØR3 R4=-R2+R4 2540 0630 to get the unreduced echelon form 0001 0000 Cols 1,2,4 have pivots. So the rank of M is 3. how the rank is limited by the size of the matrix IfAis7≈4then its rank is either 0 (if it's the zero matrix), 1, 2, 3 or 4. The rank can't be 5 or larger because there can't be 5 ind cols when there are only 4 cols to begin with. -
Irreducibility in Algebraic Groups and Regular Unipotent Elements
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 141, Number 1, January 2013, Pages 13–28 S 0002-9939(2012)11898-2 Article electronically published on August 16, 2012 IRREDUCIBILITY IN ALGEBRAIC GROUPS AND REGULAR UNIPOTENT ELEMENTS DONNA TESTERMAN AND ALEXANDRE ZALESSKI (Communicated by Pham Huu Tiep) Abstract. We study (connected) reductive subgroups G of a reductive alge- braic group H,whereG contains a regular unipotent element of H.Themain result states that G cannot lie in a proper parabolic subgroup of H. This result is new even in the classical case H =SL(n, F ), the special linear group over an algebraically closed field, where a regular unipotent element is one whose Jor- dan normal form consists of a single block. In previous work, Saxl and Seitz (1997) determined the maximal closed positive-dimensional (not necessarily connected) subgroups of simple algebraic groups containing regular unipotent elements. Combining their work with our main result, we classify all reductive subgroups of a simple algebraic group H which contain a regular unipotent element. 1. Introduction Let H be a reductive linear algebraic group defined over an algebraically closed field F . Throughout this text ‘reductive’ will mean ‘connected reductive’. A unipo- tent element u ∈ H is said to be regular if the dimension of its centralizer CH (u) coincides with the rank of H (or, equivalently, u is contained in a unique Borel subgroup of H). Regular unipotent elements of a reductive algebraic group exist in all characteristics (see [22]) and form a single conjugacy class. These play an important role in the general theory of algebraic groups. -
Rotation Matrix - Wikipedia, the Free Encyclopedia Page 1 of 22
Rotation matrix - Wikipedia, the free encyclopedia Page 1 of 22 Rotation matrix From Wikipedia, the free encyclopedia In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix rotates points in the xy -Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system. To perform the rotation, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv (see below for details). In two and three dimensions, rotation matrices are among the simplest algebraic descriptions of rotations, and are used extensively for computations in geometry, physics, and computer graphics. Though most applications involve rotations in two or three dimensions, rotation matrices can be defined for n-dimensional space. Rotation matrices are always square, with real entries. Algebraically, a rotation matrix in n-dimensions is a n × n special orthogonal matrix, i.e. an orthogonal matrix whose determinant is 1: . The set of all rotation matrices forms a group, known as the rotation group or the special orthogonal group. It is a subset of the orthogonal group, which includes reflections and consists of all orthogonal matrices with determinant 1 or -1, and of the special linear group, which includes all volume-preserving transformations and consists of matrices with determinant 1. Contents 1 Rotations in two dimensions 1.1 Non-standard orientation -
§9.2 Orthogonal Matrices and Similarity Transformations
n×n Thm: Suppose matrix Q 2 R is orthogonal. Then −1 T I Q is invertible with Q = Q . n T T I For any x; y 2 R ,(Q x) (Q y) = x y. n I For any x 2 R , kQ xk2 = kxk2. Ex 0 1 1 1 1 1 2 2 2 2 B C B 1 1 1 1 C B − 2 2 − 2 2 C B C T H = B C ; H H = I : B C B − 1 − 1 1 1 C B 2 2 2 2 C @ A 1 1 1 1 2 − 2 − 2 2 x9.2 Orthogonal Matrices and Similarity Transformations n×n Def: A matrix Q 2 R is said to be orthogonal if its columns n (1) (2) (n)o n q ; q ; ··· ; q form an orthonormal set in R . Ex 0 1 1 1 1 1 2 2 2 2 B C B 1 1 1 1 C B − 2 2 − 2 2 C B C T H = B C ; H H = I : B C B − 1 − 1 1 1 C B 2 2 2 2 C @ A 1 1 1 1 2 − 2 − 2 2 x9.2 Orthogonal Matrices and Similarity Transformations n×n Def: A matrix Q 2 R is said to be orthogonal if its columns n (1) (2) (n)o n q ; q ; ··· ; q form an orthonormal set in R . n×n Thm: Suppose matrix Q 2 R is orthogonal. Then −1 T I Q is invertible with Q = Q . n T T I For any x; y 2 R ,(Q x) (Q y) = x y. -
The Jordan Canonical Forms of Complex Orthogonal and Skew-Symmetric Matrices Roger A
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Linear Algebra and its Applications 302–303 (1999) 411–421 www.elsevier.com/locate/laa The Jordan Canonical Forms of complex orthogonal and skew-symmetric matrices Roger A. Horn a,∗, Dennis I. Merino b aDepartment of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112–0090, USA bDepartment of Mathematics, Southeastern Louisiana University, Hammond, LA 70402-0687, USA Received 25 August 1999; accepted 3 September 1999 Submitted by B. Cain Dedicated to Hans Schneider Abstract We study the Jordan Canonical Forms of complex orthogonal and skew-symmetric matrices, and consider some related results. © 1999 Elsevier Science Inc. All rights reserved. Keywords: Canonical form; Complex orthogonal matrix; Complex skew-symmetric matrix 1. Introduction and notation Every square complex matrix A is similar to its transpose, AT ([2, Section 3.2.3] or [1, Chapter XI, Theorem 5]), and the similarity class of the n-by-n complex symmetric matrices is all of Mn [2, Theorem 4.4.9], the set of n-by-n complex matrices. However, other natural similarity classes of matrices are non-trivial and can be characterized by simple conditions involving the Jordan Canonical Form. For example, A is similar to its complex conjugate, A (and hence also to its T adjoint, A∗ = A ), if and only if A is similar to a real matrix [2, Theorem 4.1.7]; the Jordan Canonical Form of such a matrix can contain only Jordan blocks with real eigenvalues and pairs of Jordan blocks of the form Jk(λ) ⊕ Jk(λ) for non-real λ.We denote by Jk(λ) the standard upper triangular k-by-k Jordan block with eigenvalue ∗ Corresponding author. -
Fifth Lecture
Lecture # 3 Orthogonal Matrices and Matrix Norms We repeat the definition an orthogonal set and orthornormal set. n Definition 1 A set of k vectors fu1; u2;:::; ukg, where each ui 2 R , is said to be an orthogonal with respect to the inner product (·; ·) if (ui; uj) = 0 for i =6 j. The set is said to be orthonormal if it is orthogonal and (ui; ui) = 1 for i = 1; 2; : : : ; k The definition of an orthogonal matrix is related to the definition for vectors, but with a subtle difference. n×k Definition 2 The matrix U = (u1; u2;:::; uk) 2 R whose columns form an orthonormal set is said to be left orthogonal. If k = n, that is, U is square, then U is said to be an orthogonal matrix. Note that the columns of (left) orthogonal matrices are orthonormal, not merely orthogonal. Square complex matrices whose columns form an orthonormal set are called unitary. Example 1 Here are some common 2 × 2 orthogonal matrices ! 1 0 U = 0 1 ! p 1 −1 U = 0:5 1 1 ! 0:8 0:6 U = 0:6 −0:8 ! cos θ sin θ U = − sin θ cos θ Let x 2 Rn then k k2 T Ux 2 = (Ux) (Ux) = xT U T Ux = xT x k k2 = x 2 1 So kUxk2 = kxk2: This property is called orthogonal invariance, it is an important and useful property of the two norm and orthogonal transformations. That is, orthog- onal transformations DO NOT AFFECT the two-norm, there is no compa- rable property for the one-norm or 1-norm. -
Advanced Linear Algebra (MA251) Lecture Notes Contents
Algebra I – Advanced Linear Algebra (MA251) Lecture Notes Derek Holt and Dmitriy Rumynin year 2009 (revised at the end) Contents 1 Review of Some Linear Algebra 3 1.1 The matrix of a linear map with respect to a fixed basis . ........ 3 1.2 Changeofbasis................................... 4 2 The Jordan Canonical Form 4 2.1 Introduction.................................... 4 2.2 TheCayley-Hamiltontheorem . ... 6 2.3 Theminimalpolynomial. 7 2.4 JordanchainsandJordanblocks . .... 9 2.5 Jordan bases and the Jordan canonical form . ....... 10 2.6 The JCF when n =2and3 ............................ 11 2.7 Thegeneralcase .................................. 14 2.8 Examples ...................................... 15 2.9 Proof of Theorem 2.9 (non-examinable) . ...... 16 2.10 Applications to difference equations . ........ 17 2.11 Functions of matrices and applications to differential equations . 19 3 Bilinear Maps and Quadratic Forms 21 3.1 Bilinearmaps:definitions . 21 3.2 Bilinearmaps:changeofbasis . 22 3.3 Quadraticforms: introduction. ...... 22 3.4 Quadraticforms: definitions. ..... 25 3.5 Change of variable under the general linear group . .......... 26 3.6 Change of variable under the orthogonal group . ........ 29 3.7 Applications of quadratic forms to geometry . ......... 33 3.7.1 Reduction of the general second degree equation . ....... 33 3.7.2 The case n =2............................... 34 3.7.3 The case n =3............................... 34 1 3.8 Unitary, hermitian and normal matrices . ....... 35 3.9 Applications to quantum mechanics . ..... 41 4 Finitely Generated Abelian Groups 44 4.1 Definitions...................................... 44 4.2 Subgroups,cosetsandquotientgroups . ....... 45 4.3 Homomorphisms and the first isomorphism theorem . ....... 48 4.4 Freeabeliangroups............................... 50 4.5 Unimodular elementary row and column operations and the Smith normal formforintegralmatrices . -
Linear Algebra with Exercises B
Linear Algebra with Exercises B Fall 2017 Kyoto University Ivan Ip These notes summarize the definitions, theorems and some examples discussed in class. Please refer to the class notes and reference books for proofs and more in-depth discussions. Contents 1 Abstract Vector Spaces 1 1.1 Vector Spaces . .1 1.2 Subspaces . .3 1.3 Linearly Independent Sets . .4 1.4 Bases . .5 1.5 Dimensions . .7 1.6 Intersections, Sums and Direct Sums . .9 2 Linear Transformations and Matrices 11 2.1 Linear Transformations . 11 2.2 Injection, Surjection and Isomorphism . 13 2.3 Rank . 14 2.4 Change of Basis . 15 3 Euclidean Space 17 3.1 Inner Product . 17 3.2 Orthogonal Basis . 20 3.3 Orthogonal Projection . 21 i 3.4 Orthogonal Matrix . 24 3.5 Gram-Schmidt Process . 25 3.6 Least Square Approximation . 28 4 Eigenvectors and Eigenvalues 31 4.1 Eigenvectors . 31 4.2 Determinants . 33 4.3 Characteristic polynomial . 36 4.4 Similarity . 38 5 Diagonalization 41 5.1 Diagonalization . 41 5.2 Symmetric Matrices . 44 5.3 Minimal Polynomials . 46 5.4 Jordan Canonical Form . 48 5.5 Positive definite matrix (Optional) . 52 5.6 Singular Value Decomposition (Optional) . 54 A Complex Matrix 59 ii Introduction Real life problems are hard. Linear Algebra is easy (in the mathematical sense). We make linear approximations to real life problems, and reduce the problems to systems of linear equations where we can then use the techniques from Linear Algebra to solve for approximate solutions. Linear Algebra also gives new insights and tools to the original problems.