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Characteristic Polynomials and Eigenvalues for Certain Classes of Pentadiagonal Matrices

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Characteristic Polynomials and Eigenvalues for Certain Classes of Pentadiagonal Matrices mathematics Article Characteristic Polynomials and Eigenvalues for Certain Classes of Pentadiagonal Matrices María Alejandra Alvarez 1,* , André Ebling Brondani 2 , Francisca Andrea Macedo França 2 and Luis A. Medina C. 1 1 Departamento de Matemáticas, Facultad de Ciencias Básicas, Universidad de Antofagasta, Antofagasta 1240000, Chile; [email protected] 2 Departamento de Matemática, Instituto de Ciências Exatas, Universidade Federal Fluminense, Volta Redonda 27213-145, RJ, Brazil; [email protected] (A.E.B.); [email protected] (F.A.M.F.) * Correspondence: [email protected] Received: 13 May 2020; Accepted: 28 June 2020; Published: 1 July 2020 Abstract: There exist pentadiagonal matrices which are diagonally similar to symmetric matrices. In this work we describe explicitly the diagonal matrix that gives this transformation for certain pentadiagonal matrices. We also consider particular classes of pentadiagonal matrices and obtain recursive formulas for the characteristic polynomial and explicit formulas for their eigenvalues. Keywords: pentadiagonal matrix; symmetric pentadiagonal matrix; eigenvalue; characteristic polynomial MSC: 15B99; 15A18 1. Introduction Tridiagonal and pentadiagonal matrices appear in several areas of mathematics and engineering, specially involving linear systems of differential equations. In [1], the authors give necessary and sufficient conditions for a matrix to be diagonally similar to a symmetric matrix. For tridiagonal matrices, the explicit construction of the diagonal matrix involved in this similar transformation is given in [2]. There are several results concerning different types of tridiagonal matrices and the obtaining of their eigenvalues and eigenvectors, see for instance [3] and [4] and the references therein. In the case of pentadiagonal matrices, there are many articles concerning algorithms for solving systems of equations associated with them. Among these works, we mention [5–12]. Furthermore, there are some results for particular cases of pentadiagonal matrices. In [13], the author gives a recurrence formula for the determinant of pentadiagonal matrices A = (Ai,j), such that Ai,j 6= 0 for ji − jj = 1. In [14], an algorithm is given to find the determinant of pentadiagonal matrices satisfying Ai,i+2 6= 0. In [15], the author shows that the characteristic polynomial for such matrices is the product of two polynomials given in terms of Chebyshev polynomials. In [16], the authors study pentadiagonal Toeplitz matrices and give determinantal identities for the symmetric and skew-symmetric cases. In [17] and [18], banded Toeplitz matrices are studied and, in particular, results on particular banded pentadiagonal Toeplitz matrices are obtained. Since every 3 × 3 matrix is a pentadiagonal one, it is clear that not every pentadiagonal matrix is similar to a symmetric matrix. In this work, we consider two classes of pentadiagonal matrices and obtain recursive formulas for the characteristic polynomials and explicit formulas for the eigenvalues of these classes of pentadiagonal matrices. The paper is organized as follows: in Section2, we present two classes of pentadiagonal matrices and show explicitly that these matrices are similar to symmetric pentadiagonal ones. In Section3 we obtain recursive formulas for the characteristic polynomials of this Mathematics 2020, 8, 1056; doi:10.3390/math8071056 www.mdpi.com/journal/mathematics Mathematics 2020, 8, 1056 2 of 12 type of matrices and a result regarding their eigenvalues. In Section4, we consider special subclasses of pentadiagonal matrices and, using the results of Section3, we show their eigenvalues and provide some results on the nullity and spectral radius of these matrices. Let Mn(R) be the set of square matrices of order n with real entries and let S ⊂ Mn(R) be the set of symmetric matrices. We denote the (i, j) entry of a matrix A by Ai,j, and in some cases, to avoid confusion, we will use the notation [A]i,j. The highest integer lower than or equal to x will be denoted by bxc. A matrix A 2 Mn(R) is called a pentadiagonal matrix if Ai,j = 0 whenever ji − jj > 2. The class of pentadiagonal matrices, denoted by Pn, consists of matrices of the form 0 1 a1 b1 d1 0 . 0 B . C Bc a b d .. C B 1 2 2 2 C B . C B .. .. .. C Be1 c2 0 C A = B C . B .. .. .. .. C B 0 . dn−2C B C B . .. .. .. .. C @ . bn−1A 0 . 0 en−2 cn−1 an We will distinguish two subclasses of pentadiagonal matrices, namely: n − 1 C = A 2 Pn : b = c = d − = e − = 0, for i = 1, 2, . , 1 2i 2i 2i 1 2i 1 2 and n j n ko C = A 2 Pn : b − = c − = d = e = 0, for i = 1, 2, . , . 2 2i 1 2i 1 2i 2i 2 For a real symmetric matrix A = (Ai,j) of order n, the non-directed graph G(A) associated to A consists of vertices f1, 2, ... , ng and edges fi, jg for which i 6= j and Ai,j Aj,i 6= 0. The matrix A is acyclic if G(A) has no cycles. 2. Pentadiagonal Matrices Similar to Symmetric Matrices In [1], the authors give necessary and sufficient conditions for a matrix A = (Ai,j) to be diagonally similar to a symmetric matrix, namely Ai,j 6= 0 implies Aj,i 6= 0, (1) and for any sequence of integers i1,..., ir such that 1 ≤ ik ≤ n, k = 1, . , r the following holds = Ai1,ir Air,ir−1 ... Ai3,i2 Ai2,i1 Ai1,i2 Ai2,i3 ... Air−1,ir Air,i1 . (2) The construction of the diagonal matrix for the case of tridiagonal matrices is done in [2]. For acyclic matrices is given in [19] and [20]. In the following theorem we impose conditions to −1 matrices A 2 C1 [C2 and give the explicit construction of a diagonal matrix D such that DAD is a symmetric pentadiagonal matrix. Theorem 1. Let A 2 C1 [C2. If any of the following conditions holds: (i)A 2 C1 satisfies bici > 0 or bi = ci = 0, for i = 1, . , n − 1; (ii)A 2 C2 satisfies diei > 0 or di = ei = 0, for i = 1, . , n − 2, then A is similar to a symmetric pentadiagonal matrix. Proof. Under assumptions (i) or (ii), A satisfies conditions (1) and (2), therefore the existence of D is guaranteed. If bici > 0 (respectively diei > 0) then the sign of bi and ci (respectively di and ei) are Mathematics 2020, 8, 1056 3 of 12 the same. Furthermore, if bi = 0 (respectively if di = 0) then ci = 0 (respectively ei = 0). We define si (respectively ti) in the following manner: 8 8 1, if b > 0; 1, if d > 0; <> i <> i si = 0, if bi = 0; and ti = 0, if di = 0; > > : −1, if bi < 0, : −1, if di < 0. We will consider each condition separately. Assume A satisfies condition (i). We will prove that A is similar to a symmetric pentadiagonal matrix Rn of order n of the form: • If n is even: p 0 1 pa1 s1 b1c1 0p 0 . 0 Bs1 b1c1 a2 0 t2 d2e2 0 C B p C B 0 0 a3 s3 b3c3 0 0 C B p p p C B 0 t2 d2e2 s3 b3c3 a 0 t d e 0 . 0 C B 4 4 4 4 C Rn = B . C , (3) B .. .. .. .. .. .. C B . C B C B . C B . p C @ . 0 0p 0 pan−1 sn−1 bn−1cn−1A 0 . 0 tn−2 dn−2en−2 sn−1 bn−1cn−1 an • If n is odd: ! Rn−1 0 Rn = , where Rn−1 is of the form (3). 0 an Let D = diagfa1, a2, ... , ang be the diagonal matrix whose diagonal entries are given, recursively, by: a1 = 1; s b2k−1 n a2k = a2k−1, for k = 1, . , 2 ; c2k−1 s r d2k c2k+1 n a2k+1 = a2k, for k = 1, . , 2 − 1; e2k b2k+1 an = 1, if n is odd. In the previous formula: s d2j−2 • If b2j−1 = c2j−1 = 0 for some j, then we replace a2j = a2j−1 = a2j−2. e2j−2 s c2j+1 • If d2j = e2j = 0 for some j, then we replace a2j+1 = a2j. b2j+1 −1 −1 Consider now X = DAD . Then X = (Xi,j) = (ai Ai,jaj ), where • If i is odd, then 8 8 a , if j = i; a , if j = i; <> i <> i p Ai,j = bi, if j = i + 1; and [Rn]i,j = si bici, if j = i + 1; :> 0, elsewhere, :> 0 elsewhere. • If i is even, then 8 8 p ei−2, if j = i − 2; ti−2 di−2ei−2, if j = i − 2; > > p > c , if j = i − 1; > s b c , if j = i − 1; <> i−1 <> i−1 i−1 i−1 Ai,j = ai, if j = i; and [Rn]i,j = ai, if j = i; > > p > di, if j = i + 2; > ti diei, if j = i + 2; > > : 0, elsewhere, : 0, elsewhere. Mathematics 2020, 8, 1056 4 of 12 Then we obtain: (a) If i is odd, then r −1 ci −1 p Xi,i = ai = [Rn]i,i, and Xi,i+1 = aibiai+1 = aibi ai = si bici = [Rn]i,i+1. bi (b) If i is even, then s s r −1 bi−1 di−2 ci−1 −1 p Xi,i−2 =aiei−2ai−2 = ai−2ei−2ai−2 = ti−2 di−2ei−2 = [Rn]i,i−2, ci−1 ei−2 bi−1 s −1 bi−1 −1 p Xi,i−1 =aici−1ai−i = ai−1ci−1ai−1 = si−1 bi−1ci−1 = [Rn]i,i−1, ci−1 −1 Xi,i =aiaiai = [Rn]i,i, r r s −1 ci+1 ei bi+1 −1 p Xi,i+2 =aidiai+2 = aidi ai = ti diei = [Rn]i,i+2.
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