Inverse of the Companion Matrix

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African Journal of Basic & Applied Sciences 10 (1): 08-09, 2018 ISSN 2079-2034 © IDOSI Publications, 2018 DOI: 10.5829/idosi.ajbas.2018.08.09

R. Cruz-Santiago, I. Guerrero-Moreno and J. López-Bonilla

ESIME-Zacatenco, Instituto Politécnico Nacional, Edif. 5, 1er. Piso, Col. Lindavista CP 07738, CDMX, México

Abstract: We use the Faddeev-Sominsky's algorithm to obtain the inverse of the companion matrix.

Key words: Companion matrix Faddeev-Sominsky's method Characteristic polynomial

INTRODUCTION therefore (1) also is the characteristic polynomial of (4) and the eigenvalues of C are identical with the proper

Here we consider an arbitrary real square matrix Anxn, values of A, besides det C = det A. whose characteristic polynomial can be obtained, for In Sec. 2 we employ the Faddeev-Sominsky's example, with the Leverrier-Takeno's technique [1]: algorithm [6-8] to construct the inverse of (4) when

an 0. nn−12 (1) +a12 + a ++... aann− 1 + =0, Inverse of the Companion Matrix: The where: Faddeev-Sominsky's procedure to obtain the inverse matrix contains the following instructions: 11 a=−= s, a [( s )23 − s ], a =−+−[ ( s ) 3 ss 2 s ], 1 1 22! 1 2 33 1 12 3 B1= C, q 1= tr B 1 , E 1= B 11 − qI , (2) 1 B= EC, q = tr B, E= B − q I , 2 1 22 2 2 22 such that:   1 r BECqn−1= n − 2, n − 1= tr BEBqI n − 1, n − 1= nn −− 11 − , sr teace A , r = 1,2,...,n (3) n −1

n in particular an = (– 1) det A (6) 1 Bnn= ECq−1 , n= tr B nn, E= 0 The companion matrix of A is given by [2-5]: qn

[Cayley-Hamilton theorem],

then:

−1 1 (4) CE= n−1, (7) qn with the property: The case n = 3 allows deduce the general structure of trace Cjj = trace A , j = 1,...,n, (5) C 1, in fact:

Corresponding Author: R. Cruz-Santiago, ESIME-Zacatenco, Instituto Politécnico Nacional, Edif. 5, 1er. Piso, Col. Lindavista CP 07738, CDMX, México. 8 African J. Basic & Appl. Sci., 10 (1): 08-09, 2018

0 1 0  a1 10    B1=0 0 1  , q1=−= aE 11 , 0 a 1 1,    −−−aaa321  −−aa320

01a1  aa211    B232=−− a a0  , q 2223 =−=− aE,  a 0 0,    0−−aa32  00−a3

−a3 00  B3=−0 a 3 0 , q 3 =−= aE 33 , 0,  00−a3

Hence:

(8)

This suggests the structure:

(9)

in harmony with the result of Wanicharpichat [9], in fact, 5. Finkbeiner, D.T., 1966. Introduction to matrices and it is simple to see that the multiplication of (4) and (9) linear transformations, W.H. Freeman, San Francisco, gives the identity matrix. USA. 6. Faddeev, D.K. and I.S. Sominsky, 1949. Collection of REFERENCES problems on higher algebra, Moscow. 7. Faddeev, D.K. and V.N. Faddeeva, 1963. Methods in 1. Guerrero-Moreno, I., J. López-Bonilla and J. linear algebra, W. H. Freeman, San Francisco, USA. Rivera-Rebolledo,2011. Leverrier-Takeno coefficients 8. López-Bonilla, J., H. Torres-Silva, S. Vidal-Beltrán, for the characteristic polynomial of a matrix, J. Inst. 2018. On the Faddeev-Sominsky's algorithm, World Eng., 8(1-2): 255-258. Scientific News, 106: 238-244. 2. Frazer, R.A., W.J. Duncan andA.R. Collar, 1963. 9. Wanicharpichat, W., 2015. Explicit minimum Elementary matrices and some applications to polynomial, eigenvector and inverse formula of dynamics and differential equations, Cambridge doubly Leslie matrix, J. Appl. Math. & Informatics University Press. 33(3-4): 247-260. 3. L. Brand, L., 1964. The companion matrix and its properties, Am. Math. Monthly, 71(6): 629-634. 4. Wilkinson, J.H., 1965. The algebraic eigenvalue problem, Clarendon Press, Oxford.