The Computation of the Inverse of a Square Polynomial Matrix


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1 The Computation of the Inverse of a Square Poynomia Matrix Ky M. Vu, PhD. AuLac Technoogies Inc. c 2008 Emai: Abstract An approach to cacuate the inverse of a square poynomia matrix is suggested. The approach consists of two simiar agorithms: One cacuates the determinant poynomia and the other cacuates the adjoint poynomia. The agorithm to cacuate the determinant poynomia gives the coefficients of the poynomia in a recursive manner from a recurrence formua. Simiary, the agorithm to cacuate the adjoint poynomia aso gives the coefficient matrices recursivey from a simiar recurrence formua. Together, they give the inverse as a ratio of the adjoint poynomia and the determinant poynomia. Keywords: Adjoint, CayeyHamiton theorem, determinant, Faddeev s agorithm, inverse matrix. 1 Introduction Modern contro theory consists of a arge part of matrix theory. A contro theorist is usuay an expert in matrix theory, a fied of appied agebra. There are books written entirey on matrix theory for contro engineers. We can cite two books: one by S. Barnett (1960) and the other by T. Kaczorek (2007). The reason for this fact is that matrices are used to represent modes of a contro system. Since matrices are used in contro theory, their manipuation and cacuation are of paramount importance for contro engineers. The most significant task is the cacuation of the inverse of a poynomia matrix. There are agorithms, discussed in textbooks, for this task. We can cite the agorithms in V. Kučera (1979) and T. Kaczorek (2007). These agorithms are poor and inefficient in the sense that they cannot be used for simiar types of poynomia matrix. In this paper, we wi present agorithms to cacuate the determinant and adjoint poynomias of an inverse poynomia matrix. The agorithms are an extension of a known agorithm by K. Vu (1999) to cacuate the characteristic coefficients of a penci. The paper is organized as foows. Section one is the introduction section. The agorithms are discussed in section two. In section three, some exampes are given. Section four concudes the discussion of the paper. A newandimproved version of this paper can be found in the Proceedings of the IEEE Muticonference on Systems and Contro on September 35th, 2008 in San Antonio, TX under the tite An Extension of the Faddeev s Agorithms. 2 The Inverse Poynomia The inverse of a square nonsinguar matrix is a square matrix, which by premutipying or postmutipying with the matrix gives an identity matrix. An inverse matrix can be expressed as a ratio of the adjoint and determinant of the matrix. A singuar matrix has no inverse because its determinant is zero; we cannot cacuate its inverse. We, however, can aways cacuate its adjoint and determinant. It is, therefore, aways better to cacuate the inverse of a square poynomia matrix by cacuating its adjoint and determinant poynomias separatey and obtain the inverse as a ratio of these poynomias. In the foowing discussion, we wi present agorithms to cacuate the adjoint and determinant poynomias. 2.1 The Determinant Poynomia To begin our discussion, we consider the foowing equation that is the determinant of a square matrix of dimension n, viz λi C λ n p 1 (C)λ + p 2 (C)λ n 2 ( 1) n, n ( 1) i p i (C)λ n i, p 0 (C) 1, i0 (λ λ 1 )(λ λ 2 ) (λ λ n ), f(λ). The coefficients p i (C) s are caed the characteristic coefficients of the matrix C. They are reated to the zeros λ i s of the poynomia f(λ) as p 1 (C) λ 1 + λ λ n, trc, p 2 (C) λ 1 λ 2 + λ 1 λ n + λ 2 λ 3 + λ 2 λ n + λ λ n,. λ 1 λ 2 λ n, C. The characteristic coefficients are given recursivey as p m (C) 1 m m ( 1) i 1 p m i (C)p 1 (C i ). i1 The coefficients p 1 (C i ) s are the Newton sums of the function f(λ). Now, consider the case C A + µb 1 + µ 2 B µ r B r. (1)
2 In this case, the function is a poynomia of degree n r in the parameter µ. Simiary, the characteristic coefficient p m (C) is a poynomia of degree m r in the parameter µ. Therefore, we can write p m (C) m r k0 q m,k (C)µ k, m 1, n. From the ast equation, we can obtain q m,k (C) 1 d k p m (C) k! dµ k, k 0, m r. Some specia vaues for these coefficients are given beow q m,0 (C) p m (A), q 0,0 (C) 1. The coefficient q m,k (C) can be cacuated recursivey, and the ast equation serves as the initia condition for the cacuation. In the foowing discussion, we wi prove this fact. From the reation of the characteristic coefficients p m (C) 1 m ( 1) i 1 p m i (C)p 1 (C i ) m i1 and the derivative of a product of two scaars d k p m i (C)p 1 (C i ) dµ k we can write q m,k (C) 1 m ( 1) i 1 k!m i1 j0 k ( k j j0 ) d k j p m i (C) dµ k j d j p 1 (C i ) dµ j, k ( ) k d k j p m i (C) d j p 1 (C i ) j dµ k j dµ j. By canceing and redistributing the factorias, we get q m,k (C) 1 m 1 m m i1 j0 m i1 j0 k ( 1) i 1 dk j p m i (C) d j p 1 (C i ) (k j)!dµ k j j!dµ j, k ( 1) i 1 q m i,k j (C) dj p 1 (C i ) j!dµ j. The derivatives of the characteristic coefficient p 1 (C i ) are not difficut to obtain, but it is compicated to obtain a genera formua for them. We, therefore, wi sove the probem as foows. By noting that we can interchange the order of the operators: trace and derivative, we write then obtain q m,k (C) 1 m d j p 1 (C i ) j!dµ j p 1 m i1 j0 ( j!dµ j ), k ( ) d ( 1) i 1 j C i q m i,k j (C) p 1 j!dµ j (2) The derivatives of the matrix C i can be cacuated recursivey. In the foowing discussion, we wi present an approach to cacuate them numericay. We write 0 C i C i 1 C, then differentiate both sides with respect to µ to obtain j ( ) j d j C i 1 d C dµ j dµ j dµ, dj C i 1 dµ j C + dj C i 2 dµ j C 2 + j 1 2 ( j j k1 1 ) d j C i 1 dµ, ( j ) d j C i k dµ Ck 1. To be abe to cacuate the derivatives numericay in a recursive manner, we have to eiminate the jth derivative of the matrix C i 2 on the right hand side of the above equation. We, therefore, wi make repeated substitutions to reduce the power of the matrix C i 2 with matrices of ower powers unti we know the jth derivative of this matrix is zero. This matrix can be an identity matrix, so we have eventuay dµ j i k1 1 j ( j ) d j C i k dµ Ck 1. The th derivative of the matrix C can be evauated as d C r dµ d dµ [A + B s µ s ], s1 r s(s 1) (s + 1)B s µ s. (3) s By putting this derivative into the previous equation, we get i j ( ) j d j C i k r dµ j dµ j s(s 1) (s + 1) k1 1 i j k1 1 j! d j C i k!(j )! dµ j s B s µ s C k 1, r s(s 1) (s +1) s B s µ s C k 1. (4) Now, by setting µ 0, Eq. (3) becomes d C r dµ s(s 1) (s + 1)B s µ s, s s!b,!b. Therefore, by setting µ 0 and dividing both sides of Eq. (4) by j!, we can cacuate the derivatives of the matrix C i recursivey as j!dµ j i k1 1 j d j C i k (j )!dµ j B A k 1
3 with the initia condition d 0 C i dµ 0 A i. With the derivatives obtained, we can take their traces and cacuate, using Eq. (2), a the coefficients q m,k (C) s of the characteristic coefficient p m (C) recursivey. The determinant corresponds to the coefficient. 2.2 The Adjoint Poynomia An agorithm, caed the Faddeev s agorithm described in D. Faddeev and I. Sominskii (1954), has been used as a standard agorithm to cacuate the determinant and adjoint poynomias of a square poynomia matrix for contro engineers. The agorithm is discussed in W. Woovich (1974) and V. Faddeeva (1959), among others. The agorithm, however, can ony be used for the resovent, [λi C] 1, of a constant matrix C. We, therefore, need an agorithm, simiar to the agorithm to cacuate the determinant poynomia, to cacuate the adjoint poynomia of the penci given by (1). To derive the equations for the agorithm, we use the reation between the characteristic coefficients of a matrix and the matrix in the CayeyHamiton theorem. From this theorem, we have C n p 1 (C)C + + ( 1) n I 0. From the above equation, we can write C 1 ( 1) i p i(c) C i, p 0 (C) 1, i0 provided that is not zero, which means that the matrix C has an inverse. We can write the ast equation as [A + µb µ r B r ] 1 This means that we have adj C C +, r() k0 1 ( 1) i p i (C)C i, i0 1 adj C, r() 1 D k µ k. D k µ k. k0 To cacuate the coefficient matrices D k s for the adjoint poynomia, we proceed as foows. We write r() j0 D k µ k ( 1) i p i (C)C i. i0 By differentiating both sides k times, dividing both sides by k! and setting µ 0, we get D k ( 1) i 1 d k k! dµ k p i(c)c i, i0 k ( 1) i 1 ( ) k d k j p i (C) k! j dµ k j dµ j, i0 j0 i0 j0 i0 j0 k ( 1) i q i,k j (C) dj C i j!dµ j, k ( 1) i q i,k j (C) dj C i j!dµ j. In this case, we do not have to take the traces of the derivative matrices, and there is a sight difference in the coefficients q m,k (C) s of the two cases. Nonetheess, once we have cacuated a these coefficients and the derivative matrices, a singe program can cacuate both the adjoint and the determinant poynomias of the inverse of a poynomia matrix. 3 Some Exampes We wi consider a verification exampe first. Suppose that we have the foowing matrices: A B 1 B With the parameter µ 0.9, we have C A + µb 1 + µ 2 B 2, The adjoint matrix of the matrix C can be cacuated directy from the above matrix, and it is obtained as C
4 The adjoint agorithm gives the foowing matrices for the adjoint poynomia, viz D D 1 D 2 D 3 D 4 D 5 D 6 It can be verified that C + D 0 + D 1 µ + D 2 µ D 6 µ 6. Simiary, we can cacuate the determinant from the matrix C. We have C The determinant agorithm gives the foowing coefficients: q 4,0 (C) , q 4,1 (C) , q 4,2 (C) , q 4,3 (C) , q 4,4 (C) , q 4,5 (C) , q 4,6 (C) , q 4,7 (C) , q 4,8 (C) It can aso be verified that C q 4,0 (C) + q 4,1 (C)µ + + q 4,8 (C)µ 8. Now, we wi consider a contro theory exampe. We wi obtain the unconstrained controer for a mutivariabe stochastic reguating contro system. The system is described by a state space mode as ŷ t+f+1 c[i Az 1 ] 1 bu t. The tripe (c, A, b) is caed a reaization of the system. The system has a pure dead time of f intervas. The input or manipuated variabe vector is u t, and the output variabe vector is ŷ t. This variabe is the undisturbed output variabe. It is disturbed by a VARIMA time series given by the foowing equation ϕ(z 1 )n t+f+1 ϑ(z 1 )a t+f+1. Then, we can write the disturbed output variabe y t of the contro system at the time (t + f + 1) as y t+f+1 ŷ t+f+1 + n t+f+1, c[i Az 1 ] 1 bu t + ϕ(z 1 ) 1 ϑ(z 1 )a t+f+1. The purpose of the controer is to produce minimum variance for y t. To obtain the controer, we wi reason as foows. At the time t when y t is avaiabe, we want to move the contro variabe u t such that the output variabe y t+f+1 0. This requires the knowedge of n t+f+1. At the time t, however, the variabe n t+f+1 is not avaiabe, so we must repace it by its minimum variance predictor. We must obtain the (f + 1) steps ahead forecast for n t. This can be done as foows. From the equation of the VARIMA time series, we have n t+f+1 ϕ(z 1 ) 1 ϑ(z 1 )a t+f+1, ϕ(z 1 ) + ϕ(z 1 ) ϑ(z 1 )a t+f+1, [ψ(z 1 ) + γ(z 1 ) ϕ(z 1 ) z f 1 ]a t+f+1, ψ(z 1 )a t+f+1 + γ(z 1 ) By taking the conditiona expectation at the time t, we obtain ˆn t+f+1 t γ(z 1 ) The controer can be obtained by setting c[i Az 1 ] 1 bu t ˆn t+f+1 t, γ(z 1 ) To obtain the controer in terms of the input and output variabes, we have to repace the white noise a t by the output variabe y t. This can be done by observing that under the minimum variance contro aw, the output variabe equas the prediction error, i.e. we have y t ψ(z 1 )a t.
5 Therefore, we have The desired controer is a t ψ(z 1 ) 1 y t, ψ(z 1 ) + ψ(z 1 ) y t. ϕ(z 1 ) ψ(z 1 ) c[i Az 1 ] + bu t I Az 1 γ(z 1 )ψ(z 1 ) + y t. In the above equation, the poynomias [I Az 1 ] + and I Az 1 can be obtained by the Faddeev s agorithm; the other poynomias, however, must be obtained by the agorithms in this paper. 4 Concusion In this paper, we have presented agorithms to cacuate the adjoint and determinant poynomias of a square poynomia matrix. The agorithms are sufficient and eegant, and they can be extended further for matrices with poynomias in two or more parameters. With such a strength, the agorithms shoud be standard agorithms, used in contro appications, for the cacuation of the adjoint and determinant poynomias. References S. Barnett (1960). Matrices in Contro Theory. Van Nostrand Reinhod Company, London. V.N. Faddeeva (1959). Computationa Methods of Linear Agebra. Dover Pubications Inc., New York. D.K. Faddeev and I.S. Sominskii (1954). Probems in Higher Agebra. Gostekhizdat, Moscow, 2nd ed. 1949, 5th ed.. T. Kaczorek (2007). Poynomia and Rationa Matrices: Appications in Dynamica Systems Theory. Springer Verag London Limited, London. V. Kučera (1979). Discrete Linear Contro: The Poynomia Equation Approach. John Wiey and Son Ltd., New York. Ky M. Vu (1999). Penci Characteristic Coefficients and Their Appications in Contro. IEE Proceedings: Contro Theory and Appications, 146: W.A. Woovich (1974). Linear Mutivariabe Systems. SpringerVerag New York Inc., New York.
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