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Explicit Minimum Polynomial, Eigenvector, and Inverse Formula for Nonsymmetric Arrowhead Matrix

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Explicit Minimum Polynomial, Eigenvector, and Inverse Formula for Nonsymmetric Arrowhead Matrix International Journal of Pure and Applied Mathematics Volume 108 No. 4 2016, 967-984 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: 10.12732/ijpam.v108i4.21 ijpam.eu EXPLICIT MINIMUM POLYNOMIAL, EIGENVECTOR, AND INVERSE FORMULA FOR NONSYMMETRIC ARROWHEAD MATRIX Wiwat Wanicharpichat 1Department of Mathematics Faculty of Science and 2Research Center for Academic Excellence in Mathematics Naresuan University Phitsanulok 65000, THAILAND Abstract: In this paper, we treat the eigenvalue problem for a nonsymmetric arrowhead matrix which is the general form of a symmetric arrowhead matrix. The purpose of this paper is to present explicit formula of determinant, inverse, minimum polynomial, and eigenvector formula of some nonsymmetric arrowhead matrices are presented. AMS Subject Classification: 15A09, 15A15, 15A18, 15A23, 65F15, 65F40 Key Words: arrowhead matrix, Krylov matrix, Schur complement, nonderogatory matrix 1. Introduction and Preliminaries Let R be the field of real numbers and C be the field of complex numbers and ∗ C = C \{0}. For a positive integer n, let Mn be the set of all n × n matrices over C. The set of all vectors, or n × 1 matrices over C is denoted by Cn.A n nonzero vector v ∈ C is called an eigenvector of A ∈ Mn corresponding to a Received: June 6, 2016 c 2016 Academic Publications, Ltd. Published: August 16, 2016 url: www.acadpubl.eu 968 W. Wanicharpichat scalar λ ∈ C if Av = λv, and the scalar λ is an eigenvalue of the matrix A. The set of eigenvalues of A is called as the spectrum of A and is denoted by σ(A). A matrix of the form d bn−1 ··· b2 b1 bn−1 sn−1 0 ··· 0 . A = . 0 .. .. ∈ M (R) (1) n . b . .. s 0 2 2 b 0 ··· 0 s 1 1 is called real symmetric arrowhead matrices (also known as real symmetric bordered diagonal matrices) [16, 21]. This class of matrices appears in certain symmetric inverse eigenvalue and inverse Sturm-Liouville problems, which arise in many applications, including control theory and vibration analysis [2, 5, 17]. O’Leary and Stewart [15] have presented formulas and efficient algorithms for computing eigenvalues and eigenvectors of symmetric arrowhead matrices. The eigenvalue problem for a symmetric arrowhead matrix arises in the description of radiationless transitions in isolated molecules and of oscillators vibrationally coupled with a Fermi liquid [6]. The properties of eigenvectors of arrowhead matrices were studied in [15]. In physics, symmetric arrowhead matrices have been used to describe radiationless transitions in isolated molecules [1, 15]. Wilkinson [21, pp.95-97] has given some relations between the eigenval- ues of the symmetric arrowhead matrix A and the diagonal entries si, i = 1, 2, . , n − 1. Stor et al. [20] has presented a new algorithm for solving an eigenvalue problem for a real symmetric arrowhead matrix, which algorithm computes all eigenvalues and all components of the corresponding eigenvectors with high relative accuracy in O(n2) operations. The algorithm is based on a shift-and-invert approach. Double precision is eventually needed to compute only one element of the inverse of the shifted matrix. Each eigenvalue and the corresponding eigenvector can be computed separately. Montano et al. [13] have given the characteristic polynomial of the arrow matrix defined in (1) as n−1 n−1 n−1 2 ∆B(t) = det(tIn − B) = (t − d) (t − si) − bi (t − sk). (2) i=1 i=1 k=1 Y X Yk=6 i A nonsymmetric eigensolver was discussed by Jessup in [10], about the rela- tionship between a tridiagonal matrix and a nonsymmetric arrowhead matrix, EXPLICIT MINIMUM POLYNOMIAL, EIGENVECTOR, AND... 969 and shown that a nonsymmetric arrowhead matrix is the sum of a diagonal ma- trix and a rank two nonsymmetric matrix. The characteristic polynomial and eigenvector formula of a special nonsymmetric arrowhead matrix with distinct diagonal entries was also presented. Maybee [11, pp.536-537] has studied about the nonsymmetric arrowhead matrix of order n + 1 real matrix which has the following format. −a b1 b2 ··· bn −b1 −c1 0 ··· 0 A = −b2 0 −c2 ··· 0 (3) . . .. −b 0 0 · · · −c n n where a ≥ 0, cj ≥ 0, 1 ≤ j ≤ n, and bj can be either positive or negative. This matrix is called the “bordered diagonal matrix of order n + 1,” and n n n+1 n−1 2 det(A) = (−1) a cj + (−1) bj ck. (4) j j Y=1 X=1 kY=6 j Thus det(A)=0 if 2 or more of the cj = 0. Notice that if some members bj in the matrix A is not 0, then A is a non-symmetric matrix. In this paper, we emphasis on the eigenvalue problem for a nonsymmetric arrowhead matrix of order n × n which is zero, except for its main diagonal and the first row and the first column of the form −d an−1 ··· a2 a1 −bn−1 −sn−1 0 ··· 0 . A = . 0 .. .. , (5) . −b . .. −s 0 2 2 −b 0 ··· 0 −s 1 1 ∗ where d, ai, bi, si ∈ C , 1 ≤ i ≤ n − 1. We will call A as a nonsymmetric arrowhead matrix. For the sake of convenience, we write the nonsymmetric arrowhead matrix A in a partitioned form as the following: −d pT A = , (6) −q −Λ (n,n) 970 W. Wanicharpichat where an−1 bn−1 . p = . and q = . ∈ Cn−1, a b 2 2 a b 1 1 ∗ where d, ai, bi, si ∈ C , 1 ≤ i ≤ n − 1 and Λ = diag(sn−1, sn−2, . , s1) is a diagonal matrix of order n − 1. Definition 1. [9, Definition 1.3.1]. A matrix B ∈ Mn is said to be similar to a matrix A ∈ Mn if there exists a nonsingular matrix S ∈ Mn such that B = S−1AS. Let 0 0 ··· 0 1 0 0 ··· 1 0 . J = . .. 0 1 ··· 0 0 1 0 ··· 0 0 (n,n) be a permutation matrix (the “backward identity matrix”) of size n × n satis- fying J = J −1 = J T . We have −s1 0 ··· 0 −b1 . 0 −s .. −b 2 2 −D −b JAJ = . =: . .. .. 0 . aT −d (n,n) 0 ··· 0 −s −b n−1 n−1 a a ··· a −d 1 2 n−1 = A, (7) where e a1 b1 a b 2 2 n− a = . and b = . ∈ C 1, . a b n−1 n−1 ∗ with d, ai, bi, si ∈ C , 1 ≤ i ≤ n − 1 and D = diag(s1, s2, . , sn−1) is a diagonal matrix of order n − 1. This show that the arrowhead symmetric matrix A EXPLICIT MINIMUM POLYNOMIAL, EIGENVECTOR, AND... 971 is similar to the matrix A via matrix J. The matrix A is another form of a nonsymmetric arrowhead matrix. The purpose of this papere is to present the explicit formulae of determinant, inverse, minimum polynomial and eigenvector formula of the nonsymmetric arrowhead matrix in (6). We recall some well-known results that will be used sequel. Solomon [19, Theorem 2] asserted that the companion matrix 0 1 0 ... 0 0 0 1 ... 0 . C = . .. 0 0 0 ... 1 −c −c −c ... −c 0 1 2 n−1 n n−1 of f(t) = t + cn−1t + ··· + c1t + c0 over a ring with unity, the matrix C is similar to its transpose 0 0 ... 0 −c0 1 0 ... 0 −c1 CT = 0 1 ... 0 −c2 . . .. 0 0 ... 1 −c n−1 via an invertible symmetric matrix c1 c2 . cn−1 1 c2 . cn−1 1 0 . P = . .. .. 0 0 . (8) . c 1 .. n−1 1 0 ... 0 0 Theorem 2 ([19, Theorem 2]). Let R be a ring with 1, let a1 . , an be elements of R and let C be the companion matrix. There exists an invertible symmetric matrix P with coefficients in R such that PCP −1 = CT . (9) 972 W. Wanicharpichat For example, if n = 3 then c1 c2 1 0 1 0 0 0 1 −1 PCP = c2 1 0 0 0 1 0 1 −c2 2 1 0 0 −c0 −c1 −c2 1 −c2 c2 − c1 0 0 −c0 T = 1 0 −c1 = C . 0 1 −c2 n Theorem 3. [9, Theorem 1.4.8]. Let A, B ∈ Mn, if x ∈ C is an eigen- vector corresponding to λ ∈ σ(B) and if B is similar to A via S, then Sx is an eigenvector of A corresponding to the eigenvalue λ. Theorem 4. [9, Theorem 3.3.15]. A matrix A ∈ Mn is similar to the companion matrix of its characteristic polynomial if and only if the minimal and characteristic polynomial of A are identical. Definition 5. [12, p. 644] A matrix A ∈ Mn for which the characteristic polynomial ∆A(t) equal to the minimum polynomial mA(t) are said to be a nonderogatory matrix. 2. The Determinant of Nonsymmetric Arrowhead Matrix −d pT Let A = be the arrowhead matrix as defined in (6). The −q −Λ (n,n) matrix A is similar to A via J, as defined in (7), where −s1 0e ··· 0 −b1 . 0 −s .. −b 2 2 D −b A = . =: . . .. .. 0 . aT −d (n,n) 0 ··· 0 −s −b e n−1 n−1 a a ··· a −d 1 2 n−1 Two similar matrices have the same determinant, we have det(A) = det(A). (10) We use LU-decomposition by Hyman’s Methode in [8, p.280] to find the EXPLICIT MINIMUM POLYNOMIAL, EIGENVECTOR, AND... 973 determinant of A, that is also the determinant of A, as follows: −D −b A = eaT −d (n,n) e In−1 0 −D −b = aT (−D)−1 1 0T −d − aT (−D)−1(−b) (n,n) (n,n) =: LU.
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