DOCSLIB.ORG

Determinants

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Determinants 12 PREFACECHAPTER I DETERMINANTS The notion of a determinant appeared at the end of 17th century in works of Leibniz (1646–1716) and a Japanese mathematician, Seki Kova, also known as Takakazu (1642–1708). Leibniz did not publish the results of his studies related with determinants. The best known is his letter to l’Hospital (1693) in which Leibniz writes down the determinant condition of compatibility for a system of three linear equations in two unknowns. Leibniz particularly emphasized the usefulness of two indices when expressing the coefficients of the equations. In modern terms he actually wrote about the indices i, j in the expression xi = j aijyj. Seki arrived at the notion of a determinant while solving the problem of finding common roots of algebraic equations. In Europe, the search for common roots of algebraic equations soon also became the main trend associated with determinants. Newton, Bezout, and Euler studied this problem. Seki did not have the general notion of the derivative at his disposal, but he actually got an algebraic expression equivalent to the derivative of a polynomial. He searched for multiple roots of a polynomial f(x) as common roots of f(x) and f (x). To find common roots of polynomials f(x) and g(x) (for f and g of small degrees) Seki got determinant expressions. The main treatise by Seki was published in 1674; there applications of the method are published, rather than the method itself. He kept the main method in secret confiding only in his closest pupils. In Europe, the first publication related to determinants, due to Cramer, ap- peared in 1750. In this work Cramer gave a determinant expression for a solution of the problem of finding the conic through 5 fixed points (this problem reduces to a system of linear equations). The general theorems on determinants were proved only ad hoc when needed to solve some other problem. Therefore, the theory of determinants had been develop- ing slowly, left behind out of proportion as compared with the general development of mathematics. A systematic presentation of the theory of determinants is mainly associated with the names of Cauchy (1789–1857) and Jacobi (1804–1851). 1. Basic properties of determinants n The determinant of a square matrix A = aij 1 is the alternated sum ( 1)σa a ...a , − 1σ(1) 2σ(2) nσ(n) σ where the summation is over all permutations σ Sn. The determinant of the n n ∈ matrix A = aij 1 is denoted by det A or aij 1 .IfdetA = 0, then A is called invertible or nonsingular. | | The following properties are often used to compute determinants. The reader can easily verify (or recall) them. 1. Under the permutation of two rows of a matrix A its determinant changes the sign. In particular, if two rows of the matrix are identical, det A = 0. Typeset by -TEX AMS 1. BASIC PROPERTIES OF DETERMINANTS 13 AC 2. If A and B are square matrices, det =detA det B. 0 B · n 3. a n = ( 1)i+ja M ,whereM is the determinant of the matrix | ij|1 j=1 − ij ij ij obtained from A by crossing out the ith row and the jth column of A (the row (echelon) expansion of the determinant or, more precisely, the expansion with respect to the ith row). (To prove this formula one has to group the factors of aij,wherej =1,...,n, for a fixed i.) 4. λα1 + µβ1 a12 ... a1n α1 a12 ... a1n β1 a12 ... a1n . . = λ . + µ . ··· ··· ··· λα + µβ a ... a αn an2 ... ann β a ... a n n n2 nn n n2 nn 5. det(AB)=detA det B. 6. det(AT )=detA. 1.1. Before we start computing determinants, let us prove Cramer’s rule.It appeared already in the first published paper on determinants. Theorem (Cramer’s rule). Consider a system of linear equations x a + + x a = b (i =1,...,n), 1 i1 ··· n in i i.e., x A + + x A = B, 1 1 ··· n n n where Aj is the jth column of the matrix A = aij 1 . Then xi det(A1,...,An)=det(A1,...,B,...,A n) , where the column B is inserted instead of Ai. Proof. Since for j = i the determinant of the matrix det(A1,...,Aj,...,An), a matrix with two identical columns, vanishes, det(A1,...,B,...,An)=det(A1,..., xjAj,...,An) = xj det(A1,...,A j,...,An)=xi det(A1,...,An). If det(A1,...,An) = 0 the formula obtained can be used to find solutions of a system of linear equations. 1.2. One of the most often encountered determinants is the Vandermonde de- terminant, i.e., the determinant of the Vandermonde matrix 2 n 1 1 x1 x1 ... x1 − . V (x1,...,xn)= . = (xi xj). − 1 x x2 ···... xn 1 i>j n n n− To compute this determinant, let us subtract the (k 1)-st column multiplied by x from the kth one for k = n, n 1,...,2. The− first row takes the form 1 − 14 DETERMINANTS (1, 0, 0,...,0), i.e., the computation of the Vandermonde determinant of order n reduces to a determinant of order n 1. Factorizing each row of the new determinant by bringing out x x we get − i − 1 2 n 2 1 x2 x2 ... x1 − . V (x1,...,xn)= (xi x1) . . − i>1 1 x x2 ···... xn 2 n n n− For n =2theidentityV (x1,x2)=x2 x1 is obvious, hence, − V (x ,...,x )= (x x ). 1 n i − j i>j Many of the applications of the Vandermonde determinant are occasioned by the fact that V (x1,...,xn) = 0 if and only if there are two equal numbers among x1,...,xn. n 1 1.3. The Cauchy determinant aij 1 ,whereaij =(xi + yj)− , is slightly more difficult to compute than the Vandermonde| | determinant. Let us prove by induction that (xi xj)(yi yj) n i>j − − aij 1 = . | | (xi + yj) i,j 1 1 For a base of induction take aij 1 =(x1 + y1)− . The step of induction will be| performed| in two stages. First, let us subtract the last column from each of the preceding ones. We get 1 1 1 1 a =(x + y )− (x + y )− =(y y )(x + y )− (x + y )− for j = n. ij i j − i n n − j i n i j 1 Let us take out of each row the factors (xi + yn)− and take out of each column, except the last one, the factors y y . As a result we get the determinant b n, n − j | ij|1 where bij = aij for j = n and bin = 1. To compute this determinant, let us subtract the last row from each of the preceding ones. Taking out of each row, except the last one, the factors xn xi 1 − and out of each column, except the last one, the factors (xn + yj)− we make it possible to pass to a Cauchy determinant of lesser size. 1.4. A matrix A of the form 010... 00 001... 00 . . .. .. .. . 000 .. 10 000... 01 a0 a1 a2 ... an 2 an 1 − − is called Frobenius’ matrix or the companion matrix of the polynomial n n 1 n 2 p(λ)=λ an 1λ − an 2λ − a0. − − − − −···− With the help of the expansion with respect to the first row it is easy to verify by induction that n n 1 n 2 det(λI A)=λ an 1λ − an 2λ − a0 = p(λ). − − − − − −···− 1. BASIC PROPERTIES OF DETERMINANTS 15 1.5. Let bi,i Z, such that bk = bl if k l (mod n) be given; the matrix n ∈ ≡ aij 1 ,whereaij = bi j, is called a circulant matrix. Let ε ,...,ε be distinct− nth roots of unity; let 1 n n 1 f(x)=b0 + b1x + + bn 1x − . ··· − Let us prove that the determinant of the circulant matrix a n is equal to | ij|1 f(ε1)f(ε2) ...f(εn). It is easy to verify that for n =3wehave 11 1 b0 b2 b1 f(1) f(1) f(1) 2 2 1 ε1 ε1 b1 b0 b2 f(ε1) ε1f(ε1) ε1f(ε1) 2 2 1 ε2 ε2 b2 b1 b0 f(ε2) ε2f(ε2) ε2f(ε2) 11 1 2 = f(1)f(ε1)f(ε2) 1 ε1 ε1 . 2 1 ε2 ε2 Therefore, V (1,ε ,ε ) a 3 = f(1)f(ε )f(ε )V (1,ε ,ε ). 1 2 | ij|1 1 2 1 2 Taking into account that the Vandermonde determinant V (1,ε1,ε2) does not vanish, we have: a 3 = f(1)f(ε )f(ε ). | ij|1 1 2 The proof of the general case is similar. n 1.6. A tridiagonal matrix is a square matrix J = aij 1 ,whereaij = 0 for i j > 1. | − | Let ai = aii for i =1,...,n,letbi = ai,i+1 and ci =ai+1,i for i =1,...,n 1. Then the tridiagonal matrix takes the form − a1 b1 0 ... 000 c1 a2 b2 ... 000 .. 0 c2 a3 . 000 . . .. .. . . . 000 .. a b 0 n 2 n 2 000... c − a − b n 2 n 1 n 1 000... 0− c − a− n 1 n − To compute the determinant of this matrix we can make use of the following k recurrent relation. Let ∆0 = 1 and ∆k = aij 1 for k 1. k | | ≥ Expanding aij 1 with respect to the kth row it is easy to verify that ∆k = ak∆k 1 bk 1ck 1∆k 2 for k 2. − − − − − ≥ The recurrence relation obtained indicates, in particular, that ∆n (the determinant of J) depends not on the numbers bi, cj themselves but on their products of the form bici.
Load more

Recommended publications
  • 1111: Linear Algebra I
    1111: Linear Algebra I Dr. Vladimir Dotsenko (Vlad) Lecture 11 Dr. Vladimir Dotsenko (Vlad) 1111: Linear Algebra I Lecture 11 1 / 13 Previously on. Theorem. Let A be an n × n-matrix, and b a vector with n entries. The following statements are equivalent: (a) the homogeneous system Ax = 0 has only the trivial solution x = 0; (b) the reduced row echelon form of A is In; (c) det(A) 6= 0; (d) the matrix A is invertible; (e) the system Ax = b has exactly one solution. A very important consequence (finite dimensional Fredholm alternative): For an n × n-matrix A, the system Ax = b either has exactly one solution for every b, or has infinitely many solutions for some choices of b and no solutions for some other choices. In particular, to prove that Ax = b has solutions for every b, it is enough to prove that Ax = 0 has only the trivial solution. Dr. Vladimir Dotsenko (Vlad) 1111: Linear Algebra I Lecture 11 2 / 13 An example for the Fredholm alternative Let us consider the following question: Given some numbers in the first row, the last row, the first column, and the last column of an n × n-matrix, is it possible to fill the numbers in all the remaining slots in a way that each of them is the average of its 4 neighbours? This is the \discrete Dirichlet problem", a finite grid approximation to many foundational questions of mathematical physics. Dr. Vladimir Dotsenko (Vlad) 1111: Linear Algebra I Lecture 11 3 / 13 An example for the Fredholm alternative For instance, for n = 4 we may face the following problem: find a; b; c; d to put in the matrix 0 4 3 0 1:51 B 1 a b -1C B C @0:5 c d 2 A 2:1 4 2 1 so that 1 a = 4 (3 + 1 + b + c); 1 8b = 4 (a + 0 - 1 + d); >c = 1 (a + 0:5 + d + 4); > 4 < 1 d = 4 (b + c + 2 + 2): > > This is a system with 4:> equations and 4 unknowns.
    [Show full text]
  • Moments of Logarithmic Derivative SO
    Alvarez, E. , & Snaith, N. C. (2020). Moments of the logarithmic derivative of characteristic polynomials from SO(2N) and USp(2N). Journal of Mathematical Physics, 61, [103506 (2020)]. https://doi.org/10.1063/5.0008423, https://doi.org/10.1063/5.0008423 Peer reviewed version Link to published version (if available): 10.1063/5.0008423 10.1063/5.0008423 Link to publication record in Explore Bristol Research PDF-document This is the author accepted manuscript (AAM). The final published version (version of record) is available online via American Institute of Physics at https://aip.scitation.org/doi/10.1063/5.0008423 . Please refer to any applicable terms of use of the publisher. University of Bristol - Explore Bristol Research General rights This document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/red/research-policy/pure/user-guides/ebr-terms/ Moments of the logarithmic derivative of characteristic polynomials from SO(N) and USp(2N) E. Alvarez1, a) and N.C. Snaith1, b) School of Mathematics, University of Bristol, UK We study moments of the logarithmic derivative of characteristic polynomials of orthogonal and symplectic random matrices. In particular, we compute the asymptotics for large matrix size, N, of these moments evaluated at points which are approaching 1. This follows work of Bailey, Bettin, Blower, Conrey, Prokhorov, Rubinstein and Snaith where they compute these asymptotics in the case of unitary random matrices. a)Electronic mail: [email protected] b)Electronic mail (Corresponding author): [email protected] 1 I.
    [Show full text]
  • 1 1.1. the DFT Matrix
    FFT January 20, 2016 1 1.1. The DFT matrix. The DFT matrix. By definition, the sequence f(τ)(τ = 0; 1; 2;:::;N − 1), posesses a discrete Fourier transform F (ν)(ν = 0; 1; 2;:::;N − 1), given by − 1 NX1 F (ν) = f(τ)e−i2π(ν=N)τ : (1.1) N τ=0 Of course, this definition can be immediately rewritten in the matrix form as follows 2 3 2 3 F (1) f(1) 6 7 6 7 6 7 6 7 6 F (2) 7 1 6 f(2) 7 6 . 7 = p F 6 . 7 ; (1.2) 4 . 5 N 4 . 5 F (N − 1) f(N − 1) where the DFT (i.e., the discrete Fourier transform) matrix is defined by 2 3 1 1 1 1 · 1 6 − 7 6 1 w w2 w3 ··· wN 1 7 6 7 h i 6 1 w2 w4 w6 ··· w2(N−1) 7 1 − − 1 6 7 F = p w(k 1)(j 1) = p 6 3 6 9 ··· 3(N−1) 7 N 1≤k;j≤N N 6 1 w w w w 7 6 . 7 4 . 5 1 wN−1 w2(N−1) w3(N−1) ··· w(N−1)(N−1) (1.3) 2πi with w = e N being the primitive N-th root of unity. 1.2. The IDFT matrix. To recover N values of the function from its discrete Fourier transform we simply have to invert the DFT matrix to obtain 2 3 2 3 f(1) F (1) 6 7 6 7 6 f(2) 7 p 6 F (2) 7 6 7 −1 6 7 6 .
    [Show full text]
  • Numerically Stable Coded Matrix Computations Via Circulant and Rotation Matrix Embeddings
    1 Numerically stable coded matrix computations via circulant and rotation matrix embeddings Aditya Ramamoorthy and Li Tang Department of Electrical and Computer Engineering Iowa State University, Ames, IA 50011 fadityar,[email protected] Abstract Polynomial based methods have recently been used in several works for mitigating the effect of stragglers (slow or failed nodes) in distributed matrix computations. For a system with n worker nodes where s can be stragglers, these approaches allow for an optimal recovery threshold, whereby the intended result can be decoded as long as any (n − s) worker nodes complete their tasks. However, they suffer from serious numerical issues owing to the condition number of the corresponding real Vandermonde-structured recovery matrices; this condition number grows exponentially in n. We present a novel approach that leverages the properties of circulant permutation matrices and rotation matrices for coded matrix computation. In addition to having an optimal recovery threshold, we demonstrate an upper bound on the worst-case condition number of our recovery matrices which grows as ≈ O(ns+5:5); in the practical scenario where s is a constant, this grows polynomially in n. Our schemes leverage the well-behaved conditioning of complex Vandermonde matrices with parameters on the complex unit circle, while still working with computation over the reals. Exhaustive experimental results demonstrate that our proposed method has condition numbers that are orders of magnitude lower than prior work. I. INTRODUCTION Present day computing needs necessitate the usage of large computation clusters that regularly process huge amounts of data on a regular basis. In several of the relevant application domains such as machine learning, arXiv:1910.06515v4 [cs.IT] 9 Jun 2021 datasets are often so large that they cannot even be stored in the disk of a single server.
    [Show full text]
  • Some Results on Vandermonde Matrices with an Application to Time Series Analysis∗
    SIAM J. MATRIX ANAL. APPL. c 2003 Society for Industrial and Applied Mathematics Vol. 25, No. 1, pp. 213–223 SOME RESULTS ON VANDERMONDE MATRICES WITH AN APPLICATION TO TIME SERIES ANALYSIS∗ ANDRE´ KLEIN† AND PETER SPREIJ‡ Abstract. In this paper we study Stein equations in which the coefficient matrices are in companion form. Solutions to such equations are relatively easy to compute as soon as one knows how to invert a Vandermonde matrix (in the generic case where all eigenvalues have multiplicity one) or a confluent Vandermonde matrix (in the general case). As an application we present a way to compute the Fisher information matrix of an autoregressive moving average (ARMA) process. The computation is based on the fact that this matrix can be decomposed into blocks where each block satisfies a certain Stein equation. Key words. ARMA process, Fisher information matrix, Stein’s equation, Vandermonde matrix, confluent Vandermonde matrix AMS subject classifications. 15A09, 15A24, 62F10, 62M10, 65F05, 65F10, 93B25 PII. S0895479802402892 1. Introduction. In this paper we investigate some properties of (confluent) Vandermonde and related matrices aimed at and motivated by their application to a problem in time series analysis. Specifically, we show how to apply results on these matrices to obtain a simpler representation of the (asymptotic) Fisher information matrixof an autoregressive moving average (ARMA) process. The Fisher informa- tion matrixis prominently featured in the asymptotic analysis of estimators and in asymptotic testing theory, e.g., in the classical Cram´er–Rao bound on the variance of unbiased estimators. See [10] for general results and see [2] for time series models.
    [Show full text]
  • Zero-Sum Triangles for Involutory, Idempotent, Nilpotent and Unipotent Matrices
    Zero-Sum Triangles for Involutory, Idempotent, Nilpotent and Unipotent Matrices Pengwei Hao1, Chao Zhang2, Huahan Hao3 Abstract: In some matrix formations, factorizations and transformations, we need special matrices with some properties and we wish that such matrices should be easily and simply generated and of integers. In this paper, we propose a zero-sum rule for the recurrence relations to construct integer triangles as triangular matrices with involutory, idempotent, nilpotent and unipotent properties, especially nilpotent and unipotent matrices of index 2. With the zero-sum rule we also give the conditions for the special matrices and the generic methods for the generation of those special matrices. The generated integer triangles are mostly newly discovered, and more combinatorial identities can be found with them. Keywords: zero-sum rule, triangles of numbers, generation of special matrices, involutory, idempotent, nilpotent and unipotent matrices 1. Introduction Zero-sum was coined in early 1940s in the field of game theory and economic theory, and the term “zero-sum game” is used to describe a situation where the sum of gains made by one person or group is lost in equal amounts by another person or group, so that the net outcome is neutral, or the sum of losses and gains is zero [1]. It was later also frequently used in psychology and political science. In this work, we apply the zero-sum rule to three adjacent cells to construct integer triangles. Pascal's triangle is named after the French mathematician Blaise Pascal, but the history can be traced back in almost 2,000 years and is referred to as the Staircase of Mount Meru in India, the Khayyam triangle in Persia (Iran), Yang Hui's triangle in China, Apianus's Triangle in Germany, and Tartaglia's triangle in Italy [2].
    [Show full text]
  • Analytical Solution of Linear Ordinary Differential Equations by Differential
    ANALYTICAL SOLUTION OF LINEAR ORDINARY DIFFERENTIAL EQUATIONS BY DIFFERENTIAL TRANSFER MATRIX METHOD SINA KHORASANI AND ALI ADIBI Abstract. We report a new analytical method for exact solution of homoge- neous linear ordinary differential equations with arbitrary order and variable coefficients. The method is based on the definition of jump transfer matrices and their extension into limiting differential form. The approach reduces the nth-order differential equation to a system of n linear differential equations with unity order. The full analytical solution is then found by the pertur- bation technique. The important feature of the presented method is that it deals with the evolution of independent solutions, rather than its derivatives. We prove the validity of method by direct substitution of the solution in the original differential equation. We discuss the general properties of differential transfer matrices and present several analytical examples, showing the appli- cability of the method. We show that the Abel-Liouville-Ostogradski theorem can be easily recovered through this approach. 1. Introduction Solution of ordinary differential equations (ODEs) is in general possible by dif- ferent methods [1]. Lie group analysis [2-6] can be effectively used in cases when a symmetry algebra exists. Factorization methods are reported for reduction of ODEs into linear autonomous forms [7,8] with constant coefficients, which can be readily solved. But these methods require some symbolic transformations and are complicated for general higher order ODEs, and thus not suitable for many com- putational purposes where the form of ODE is not available in a simple closed form. arXiv:math-ph/0301010v7 12 Mar 2003 General second-order linear ODEs can be solved in terms of special function [9] and analytic continuation [10].
    [Show full text]
  • Fast Algorithms to Compute Matrix Vector Products for Pascal Matrices
    FAST ALGORITHMS TO COMPUTE MATRIX-VECTOR PRODUCTS FOR PASCAL MATRICES∗ ZHIHUI TANG† , RAMANI DURAISWAMI‡, AND NAIL GUMEROV§ Abstract. The Pascal matrix arises in a number of applications. We present a few ways to decompose the Pascal matrices of size n n into products of matrices with structure. Based on these decompositions, we propose fast algorithms× to compute the product of a Pascal matrix and a vector with complexity O(n log n). We also present a strategy to stabilize the proposed algorithms. Finally, we also present some interesting properties of the Pascal matrices that help us to compute fast the product of the inverse of a Pascal matrix and a vector, and fast algorithms for generalized Pascal Matrices. Key words. Matrix-vector product, Pascal matrices, matrix decomposition, structured matri- ces, Toeplitz matrices, Hankel matrices, Vandermonde matrices Introduction. If we arrange the binomial coefficients r! Cn = ,r=0, 1, 2, ··· ,n=0, 1, ··· ,r (0.1) r n!(r − n)! in staggered rows, 1 11 121 1331 , (0.2) 14641 1 5 10 10 5 1 1 6 15 20 15 6 1 we obtain the famous Pascal triangle. However, Pascal was not the first to discover this triangle, it had been described about 500 years earlier by Chinese mathematician Yang Hui. Therefore in China, it is known as “Yang Hui’s triangle" . It also has been discovered in India, Persia and Europe. For a fascinating introduction, see [Edwards02]. The Pascal matrix arises in many applications such as in binomial expansion, probability, combinatorics, linear algebra [Edelman03], electrical engineering [Biolkova99] and order statistics [Aggarwala2001].
    [Show full text]
  • On the Vandermonde Matrix
    On the Vandermonde Matrix Tom Copeland ([email protected]) May 2014 Part I The n-simplices and the Vandermonde matrix There are several denitions of the Vandermonde matrix Vn. Here 2 1 1 ··· 1 3 6 x1 x2 ··· xn 7 6 . 7 Vn = Vn(x1; x2; :::; xn) = 6 . .. 7 : 4 . 5 n−1 n−1 n−1 x1 x2 ··· xn with the determinant Y jVnj = jVn(x1; : : : ; xn)j = (xj − xi): 1≤i<j≤n The coordinates of the n vertices Cn−1(x1);Cn−1(x2); :::; Cn−1(xn) of the 2 n−1 (n − 1)-simplex along the unique moment curve Cn−1(x) = (x; x ; :::; x ) (ra- tional normal curve) that passes through the vertices comprise the columns of Vn (except for the rst row of ones). With one of the vertices located at the origin, say vertex m with xm = 0, the determinant gives the (possibly signed) hyper-volume of the hyper-parallelepiped framed by the position vectors from the origin to the other vertices, which bound the facet opposing the vertex at the origin (see the V 3 example below). Note that the xj are unit-less parameters, i.e., have no physical units or measures attached to them and in that sense are dimensionless. In other words, k should not be directly interpreted as a quantity with dimension a length, xj k an area, a volume, etc. It is a dimensionless parameter that must be interpreted in context. Ignoring this leads to confusion, such as believing that jV4j with six factors in its product formula must somehow have dimension six.
    [Show full text]
  • Extreme Points of the Vandermonde Determinant on the Sphere and Some Limits Involving the Generalized Vandermonde Determinant
    Extreme points of the Vandermonde determinant on the sphere and some limits involving the generalized Vandermonde determinant Karl Lundengård, Jonas Österberg, Sergei Silvestrov Mälardalen University December 24, 2013 Abstract The values of the determinant of Vandermonde matrices with real elements are analyzed both visually and analytically over the unit sphere in various di- mensions. For three dimensions some generalized Vandermonde matrices are analyzed visually. The extreme points of the ordinary Vandermonde deter- minant on finite-dimensional unit spheres are given as the roots of rescaled Hermite polynomials and a recursion relation is provided for the polynomial coefficients. Analytical expressions for these roots are also given for dimension three to seven. A transformation of the optimization problem is provided and some relations between the ordinary and generalized Vandermonde matrices involving limits are discussed. Keywords: Vandermonde matrix, Determinants, Extreme points, Unit sphere, Generalized Vandermonde matrix. arXiv:1312.6193v1 [math.CA] 21 Dec 2013 1 Contents 1 Introduction 3 1.1 Visual exploration in 3D . .4 2 Optimizing the Vandermonde determinant over the unit sphere 8 2.1 The extreme points given by roots of a polynomial . .8 2.2 Extreme points of the Vandermonde determinant on the three dimen- sional unit sphere . 15 2.3 A transformation of the optimization problem . 16 2.4 Further visual exploration . 18 2.4.1 Four dimensions . 19 2.4.2 Five dimensions . 20 2.4.3 Six dimensions . 21 2.4.4 Seven dimensions . 23 3 Some limit theorems involving the generalized Vandermonde matrix 24 4 Conclusions 27 2 1 Introduction Here a generalized Vandermonde matrix is determined by two vectors ~xn = (x1; ··· ; xn) 2 n m K and ~am = (a1; ··· ; am) 2 K , where K is usually the real or complex field, and is defined as G (~x ;~a ) = xai : (1) mn n m j mn For square matrices only one index is given, Gn ≡ Gnn.
    [Show full text]
  • Unified Canonical Forms for Matrices Over a Differential Ring
    Unified Canonical Forms for Matrices Over a Differential Ring J. Zhu and C. D. Johnson Electrical and Computer Engineering Department University of Alabama in Huntsville Huntsville, Alabama 35899 Submitted by Russell Menis ABSTRACT Linear differential equations with variable coefficients of the vector form (i) % = A(t)x and the scalar form (ii) y(“) + cy,(t)y(“-‘) + . * . + a2(thj + a,(t)y = 0 can be studied as operators on a differential module over a differential ring. Using this differential algebraic structure and a classical result on differential operator factoriza- tions developed by Cauchy and Floquet, a new (variable) eigenvalue theory and an associated set of matrix canonical forms are developed in this paper for matrices over a differential ring. In particular, eight new spectral and modal matrix canonical forms are developed that include as special cases the classical companion, Jordan (diagonal), and (generalized) Vandermonde canonical forms, as traditionally used for matrices over a number ring (field). Therefore, these new matrix canonical forms can be viewed as unifications of those classical ones. Two additional canonical matrices that perform order reductions of a Cauchy-Floquet factorization of a linear differential operator are also introduced. General and explicit formulas for obtaining the new canonical forms are derived in terms of the new (variable) eigenvalues. The results obtained here have immediate applications in diverse engineering and scientific fields, such as physics, control systems, communications, and electrical networks, in which linear differential equations (i) and (ii) are used as mathematical models. 1. INTRODUCTION This paper is concerned with the class of finite-dimensional, homoge- neous linear differential equations of the vector form S = A( t)x, x(h) = x0 (1.1) LlNEAR ALGEBRA AND ITS APPLlCATlONS 147:201-248 (1991) 201 0 Elsevier Science Publishing Co., Inc., 1991 655 Avenue of the Americas, New York, NY 10010 0024-3795/91/$3.50 202 J.
    [Show full text]
  • On the Closed Form Expression of Elementary Symmetric Polynomials and the Inverse of Vandermonde Matrix
    ON THE CLOSED FORM EXPRESSION OF ELEMENTARY SYMMETRIC POLYNOMIALS AND THE INVERSE OF VANDERMONDE MATRIX MAHDI S. HOSSEINI∗, ALFRED CHEN∗, AND KONSTANTINOS N. PLATANIOTIS∗ Abstract. Inverse Vandermonde matrix calculation is a long-standing problem to solve non- singular linear system V c = b where the rows of a square matrix V are constructed by progression of the power polynomials. It has many applications in scientific computing including interpolation, super-resolution, and construction of special matrices applied in cryptography. Despite its numerous applications, the matrix is highly ill-conditioned where specialized treatments are considered for ap- proximation such as conversion to Cauchy matrix, spectral decomposition, and algorithmic tailoring of the numerical solutions. In this paper, we propose a generalized algorithm that takes arbitrary pairwise (non-repetitive) sample nodes for solving inverse Vandermonde matrix. This is done in two steps: first, a highly balanced recursive algorithm is introduced with O(N) complexity to solve the combinatorics summation of the elementary symmetric polynomials; and second, a closed-form solu- tion is tailored for inverse Vandermonde where the matrix' elements utilize this recursive summation for the inverse calculations. The numerical stability and accuracy of the proposed inverse method is analyzed through the spectral decomposition of the Frobenius companion matrix that associates with the corresponding Vandermonde matrix. The results show significant improvement over the state-of-the-art solutions using specific nodes such as Nth roots of unity defined on the complex plane. A basic application in one dimensional interpolation problem is considered to demonstrate the utility of the proposed method for super-resolved signals.
    [Show full text]