Efficient algorithms for finding the minimal polynomials and the

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1 Efficient algorithms for finding the minimal polynomials and the inverses of level- FLS r 1 r -circulant matrices Linyi University Department of mathematics Linyi Shandong China jzh108@sina.com Abstract: The level- FLS r 1 r -circulant matrix over any field is introduced. The diagonalization and spectral decomposition of level- FLS r 1 r -circulant matrices over any field are discussed. Algorithms for computing the minimal polynomial of this ind of matrices over any field are presented by means of the algorithm for the Gröbner basis of the ideal in the polynomial ring and two algorithms for finding the inverses of such matrices are also presented. Finally an algorithm for the inverse of partitioned matrix with level- FLS r 1 r -circulant blocs over any field is given by using the Schur complement which can be realized by CoCoA 4.0 an algebraic system over the field of rational numbers or the field of residue classes of modulo prime number. Key Words: Level- FLS r 1 r -circulant matrix minimal polynomial inverse diagonalization spectral decomposition Gröbner basis. 1 Introduction With the development of the mathematics research multilevel circulant matrix had been defined. And it has been used on networ engineering approximate calculation and Image processing [1-4]. W. F. Trench [5 6] considered properties of unilevel bloc circulates and multilevel bloc α-circulates. S. Zhang Z. Jiang and S. Liu [7] gave Algorithms for the minimal polynomial and the inverse of a levelnr 1 r r n -bloc circulant matrix over any field are presented by means of the algorithm for the Gröbner basis for the ideal of the polynomial ring over the field. M. Morhac V. Matouse [8] presents an efficient algorithm to solve a one-dimensional as well as n-dimensional circulant convolution system. M. Rezghi L. Elden [9] defined tensors with diagonal and circulant structure and developed framewor for the analysis of such tensors. S. Georgiou and C. Kououvinos [10] presented a new method for constructing multilevel supersaturated designs. Z. Jiang and S. Liu [11] introduced the level-m scaled circulant factor matrix over the complex number field and discussed its diagonalization and spectral decomposition and representation. A. J. H. Bloc [1] considered the property of circulates of level-. J. Baer discussed the structure of multi-bloc circulates in [13]. More details on multilevel circulant matrix see [14 15]. This paper is devoted to study the level- FLS r 1 r -circulant matrix over any field and it is organized as follows. In Section 1 a level- FLS r 1 r -circulant matrix over any field is introduced and its algebraic properties are given. In addition the diagonalization and spectral decomposition of level- FLS r 1 r -circulant matrices over F are discussed. In Section we show that the ring of all level FLS r 1 r -circulant matrices over a field is isomorphic to a factor ring of a polynomial ring in variables over the same field and then present an algorithm for the minimal polynomial of a level- FLS r 1 r -circulant matrix by mean of the algorithm for the Gröbner basis for a ernel of a ring homomorphism. In Section 3 we give a sufficient and necessary condition to determine whether a level- FLS r 1 r -circulant matrix over a field is singular or not and then present an algorithm for finding the inverse of a level- FLS r 1 r -circulant matrix over a field. In Section 4 an algorithm for finding the inverse of partitioned matrix with level- FLS r 1 r - circulant blocs over a field is presented by using the Schur complement and the Buchberger s algorithm. We first introduce some terminologies and notation used in the sequel. Let F be a field and E-ISSN: Issue 4 Volume 1 April 013

2 F[x 1 x ] the polynomial ring in variables over a field F. By Hilbert Basis Theorem we now that every ideal I in F[x 1 x ] is finitely generated. Fixing a term order in F[x 1 x ] a set of non-zero polynomials G {g 1 g t } in an ideal I is called a Gröbner basis for I if and only if for all non-zero f I there exists i {1 t} lpg i divides lpf lpg i and lpf are the leading power products of g i and f respectively. A Gröbner basis G {g 1 g t } is called a reduced Gröbner basis if and only if for all i lcg i 1 and g i is reduced with respect to G {g i } that is for all i no non-zero term in g i is divisible by any lpg j for any j i lcg i is the leading coefficient of g i. In this paper we set A 0 I for any square matrix A and < f 1 f m > denotes an ideal of F[x 1 x ] generated by polynomials f 1 f m. Diagonalization and spectral decomposition of level- FLS r 1 r -circulant matrices We define ℵ r as the basic FLS r-circulant matrix over F that is ℵ r r n n 1 It is easily verified that the polynomial gx x n x r is both the minimal polynomial and the characteristic polynomial of the matrix ℵ r if r 0 and r n 1 1 nn 1 n. In addition ℵ n r is nonsingular nonderogatory and ℵ n r ri n + ℵ r. Let ℵ ri be basic FLS r i -circulant matrix over F and let I ni be the n i n i unit matrix for i 1 and N n 1 n n. Set Π i I n1 I ni 1 ℵ ri I ni+1 I n r i 0 and ri n 1 1 nn 1 n for i 1 n is a Kronecer product of matrices. Definition 1 An N N matrix A over F is called a level- F LS r 1 r circulant matrix if there exists a polynomial fx 1 x n 1 1 i 1 0 i 0 n 1 i 0 i 1 0 i 0 x i 1 1 x i F[x 1 x ] A fπ 1 Π 1 i 0 Π i 1 1 Π i the polynomial fx 1 x will be called the representor of a level- FLS r 1 r - circulant matrix A and the coefficients i j 1 n j j 1 are just the entries of the first row of A. Obviously if 1 then we obtain the FLS r- circulant matrix [5]. By the property of the Kronecer product of matrices the level- FLS r 1 r - circulant matrix A can be also expressed as A i 1 0 i 0 1 i 0 ℵ i 1 r1 ℵ i r ℵ i r. For matrix A over F if i1 r i 0 then A is a level FLS r 1 r -circulant matrix if and only if A commutes with the ℵ r1 ℵ r ℵ r that is Aℵ r1 ℵ r ℵ r ℵ r1 ℵ r ℵ r A. In addition to the algebraic properties that can be easily derived from the representation we mention that level- FLS r 1 r -circulant matrices have very nice structure. The product of two level- FLS r 1 r -circulant matrices is also a level- FLS r 1 r -circulant matrix. Furthermore level- FLS r 1 r -circulant matrices commute under multiplication and A 1 is a level- FLS r 1 r - circulant matrix too. The following we consider diagonalization and spectral decomposition of level- FLS r 1 r - circulant matrices over F. In this section let ε i0 ε i1 ε ini 1 be n i distinct roots of g i x i x n i i x i r i in its splitting field over F and V ri V ε i0 ε i1 ε ini 1 denotes the Vandermonde matrix of the ε ij s. Then Vr 1 i ℵ ri V ri D gi diagε i0 ε i1 ε ini 1 i 1 3 E-ISSN: Issue 4 Volume 1 April 013

3 Theorem Let A fπ 1 Π be a level- FLS r 1 r -circulant matrix over F fx 1 x n 1 1 Then i 1 0 i 0 n 1 i 0 x i 1 1 x i F[x 1 x ]. V r1 V r V r 1 AV r1 V r V r fd g1 D g D g diagfε 10 ε 0 ε 0 fε 1n ε n ε n 1. Proof: By equations 3 and the property of Kronecer product of matrices we have V r1 V r V r 1 fπ 1 Π V r1 V r V r V r1 V r V r 1 n 1 1 n 1 Π i 1 1 Π i i 1 0 i 0 V r1 V r V r n 1 1 i 1 0 n 1 i 0 [V r1 V r 1 ℵ i 1 r1 ℵ i r V r1 V r ] n 1 1 i 1 0 n 1 V 1 r n 1 1 i 1 0 i 0 ℵ i n 1 i 0 V 1 r 1 ℵ i 1 r1 V r1 V 1 r ℵ i r V r r V r fd g1 D g D g D i 1 g1 D i g D i g diagfε 10 ε 0 ε 0 fε 1n ε n ε n 1. Corollary 3 Let A fπ 1 Π be a level- FLS r 1 r -circulant matrix over F. Then a A is a level- FLS r 1 r -circulant matrix over F if and only if V r1 V r V r 1 AV r1 V r V r is a diagonal matrix. b The eigenvalues of A are given by λ j1 j...j i 1 0 i 0 1 i 0 ε i 1 1j1 ε i j ε i j j l 0 1 n l 1 l 1. By equation 3 the basic FLS r i -circulant matrix ℵ ri over F is written as We now let ℵ ri V ri D gi V 1 r i i 1. Ω ji diag[ ] 0 1 n i 1 i 1 1 occupies the + 1th entry. Then B r i ℵ ri i 1 0 ε iji B r i i 1 V ri Ω ji V 1 r i 0 1 n i 1 i 1 and then {B r i that is i Br i B r i 1 Br i n i 1 } is the spectral basis I ni B r i B r i l i δ ji l i B r i l i 0 1 n i 1 i 1. Moreover one can easily express the basis {I ni ℵ ri ℵ n i 1 r i } in terms of the basis {B r i 0 Br i 1 Br i n i 1 } by ℵ s r i i 1 0 ε s i B r i i 1. Furthermore {B r 1r...r j 1 j...j 0 1 n i 1 i 1 } is the spectral basis that is j 1 0 j 0 1 j 0 B r 1r...r j 1 j...j I n1 n...n B r 1r...r j 1 j...j B r 1r...r l 1 l...l δ j1 j...j l 1 l...l B r 1r...r j 1 j...j and l i 0 1 n i 1 i 1 B r 1r...r j 1 j...j B r 1 j 1 B r j B r j V r1 r...r Ω j1 Ω j Ω j V 1 r 1 r...r V r1 r...r V r1 V r V r. We summarize our discussion in the following. E-ISSN: Issue 4 Volume 1 April 013

4 Theorem 4 A level- FLS r 1 r -circulant matrix over F can be represented as A i 1 0 i 0 i 1 0 i 0 1 i 0 1 i 0 Π i 1 1 Π i ℵ i 1 r1 ℵ i r ℵ i r V 1 r 1 r...r fd g1 D g D g V r1 r...r λ j1 j...j and j 1 0 j 0 1 j 0 λ j1 j...j B r 1r...r j 1 j...j V r1 r...r V r1 V r V r i 1 0 i 0 1 i 0 ε i 1 1j1 ε i j ε i j B r 1r...r j 1 j...j B r 1 j 1 B r j B r j V r1 r...r Ω j1 Ω j Ω j V 1 r 1 r...r. 3 Efficient algorithms for finding the minimal polynomials of level- FLS r 1 r -circulant matrices Let F[Π 1 Π ] {A A fπ 1 Π fx 1 x F[x 1 x ]}. It is a routine to prove that F[Π 1 Π ] is a commutative ring with the matrix addition and multiplication. Theorem 5 F[x 1 x ]/ x n 1 1 x 1 r 1 x n x r F[Π 1 Π ]. Proof: Consider the following F -algebra homomorphism for φ : F[x 1 x ] F[Π 1 Π ] fx 1 x A fπ 1 Π fx 1 x F[x 1 x ]. It is clear that φ is an F -algebra epimorphism. So we have F[x 1 x ]/ er φ F[Π 1 Π ]. We can prove that er φ x n 1 In fact for i 1 because Hence x n i i x i r i erφ Π n i i Π i r i 0. er φ x n 1 Conversely for any fx 1 x er φ we have A fπ 1 Π 0. Fix the lexicographical order on F[x 1 x ] with x 1 > x > > x. x n 1 1 x 1 r 1 dividing fx 1 x there exist u 1 x 1 x v 1 x 1 x F[x 1 x ] fx 1 x u 1 x 1 x x n 1 1 x 1 r 1 +v 1 x 1 x v 1 x 1 x 0 or the largest degree of x 1 in v 1 x 1 x is less than n 1. If v 1 x 1 x 0 then fx 1 x x n 1 Otherwise x n x r dividing v 1 x 1 x there exist u x 1 x v x 1 x F[x 1 x ] v 1 x 1 x u x 1 x x n x r +v x 1 x v x 1 x 0 or the largest degree of x in v x 1 x is less than n. If v x 1 x 0 then fx 1 x x n 1 Otherwise if the largest degree of x 1 in v x 1 x is less than n 1 because x 1 does E-ISSN: Issue 4 Volume 1 April 013

5 not appear in x n x r. Continuing this procedure there exist u 1 x 1 x u x 1 x u x 1 x v x 1 x F[x 1 x ] fx 1 x u 1 x 1 x x n 1 1 x 1 r 1 + +u x 1 x x n x r +v x 1 x v x 1 x 0 or the degrees of x 1 x x in v x 1 x are less than n 1 n n respectively. Since fπ 1 Π 0 and Π n i i Π i r i 0 for all i 1 u Π 1 Π 0. The coefficients of all terms in v x 1 x are the entries of the matrix v Π 1 Π because the degrees of x 1 x x in v x 1 x are less than n 1 n n respectively. Therefore the coefficient of each term in v x 1 x is 0 i.e. v x 1 x 0. Thus fx 1 x x n 1 Definition 6 Let I be a non-zero ideal of the polynomial ring F[y 1 y t ]. Then I is called an annihilation ideal of square matrices A 1 A t denoted by IA 1 A t if fa 1 A t 0 for all fy 1 y t I. Definition 7 Suppose that A 1 A t F[Π 1 Π ] are not all zero matrices. The unique monic polynomial gx of minimum degree that simultaneously annihilates A 1 A t is called the common minimal polynomial of A 1 A t. We give the special case of [16 Theorem.4.10] here for the convenience of applications. Lemma 8 Let I be an ideal of F[x 1 x ]. Given f 1 f m F[x 1 x ] consider the following F - algebra homomorphism φ : F[y 1 y m ] F[x 1 x ]/I y 1 f 1 + I y m f m + I Let K I y 1 f 1 y m f m be an ideal of F[x 1 x y 1 y m ] generated by I y 1 f 1 y m f m. Then er φ K F[y 1 y m ]. Lemma 9 [19] Let A be a non-zero matrix over F if the minimal polynomial of A is: px a 0 x n + a 1 x n 1 + a x n + + a n and a n 0 then A 1 1 a n a 0 A n 1 a 1 A n a n 1 I. The following Lemma is the Exercise.38 of [16]. Lemma 10 Let L 1 L L m be ideals of F[x 1 x x ] and let J 1 m w i w 1 L 1 w L w m L m i1 be an ideal of F[x 1 x x w 1 w m ] generated by 1 m i1 w i w 1 L 1 w L w m L m. Then m L i J F[x 1 x x ]. i1 By the Theorem 5 and the Lemma 8 we can prove the following theorem. Theorem 11 The minimal polynomial of a level- FLS r 1 r -circulant matrix A F[Π 1 Π ] is the monic polynomial that generates the ideal x n 1 1 x 1 r 1 x n x r y fx 1 x F[y] the polynomial fx 1 x is the representer of A. Proof: Consider the following F - algebra homomorphism ϕ : F[y] F[x 1 x ]/ x n 1 1 x 1 r 1 x n x r F[Π 1 Π ] y fx 1 x + x n 1 x r A fπ 1 Π. It is clear that qy er ϕ if and only if qa 0. By Lemma 8 we have er ϕ x n 1 x r y fx 1 x F[y]. By Theorem 11 and Lemma 9 we now that the minimal polynomial and the inverse of a level- E-ISSN: Issue 4 Volume 1 April 013

6 FLS r 1 r -circulant matrix A F[Π 1 Π ] is calculated by a Gröbner basis for a ernel of an F -algebra homomorphism. Therefore we have the following algorithm to calculate the minimal polynomial and the inverse of a level- FLS r 1 r -circulant matrix A fπ 1 Π : Step 1 Calculate the reduced Gröbner basis G for the ideal x n 1 x r y fx 1 x F[y] by CoCoA 4.0 using an elimination order with x 1 > x > > x > y. Step Find the polynomial in G in which the variables x 1 x x are not appear. This polynomial px is the minimal polynomial of A. Step 3 By step if a n in the minimal polynomial of A px a 0 x n + a 1 x n 1 + a x n + + a n is zero stop. Otherwise calculate A 1 1 a n a 0 A n 1 a 1 A n a n 1 I. Example 1 Let A fπ 1 Π be a level- FLS 5-circulant matrix fx y x 3 y + 3x 3 y + x y + 7x 3 +x y + x + 3xy + 4y + 5xy + x + 3y +. and Π 1 ℵ I 3 Π I 4 ℵ 5 and ℵ I I 4 ℵ We can now calculate the minimal polynomial and the inverse of A with coefficients in the field Z 11 as following: In fact the reduced Gröbner basis for the ideal x 4 x y 3 y 5 z fx y : G {z 1 4z z 10 z 9 5z 8 + 4z 6 +z 5 + 3z 4 + 5z 3 + 3z 3z + 1 x + z z 9 5z 8 4z 7 + 5z 6 5z 5 + 4z 3 z + 3z + 1 y z 11 + z 10 3z 9 z 7 z 6 z 5 4z 4 z 3 z }. So the minimal polynomial of A is z 1 4z z 10 z 9 5z 8 + 4z 6 + z 5 and the inverse of A is +3z 4 + 5z 3 + 3z 3z + 1 A 1 A A 10 3A 9 + A 8 + 5A 7 4A 5 A 4 3A 3 5A 3A + 3I. Theorem 13 The annihilation ideal of level- FLS r 1 r -circulant matrices A 1 A t F[Π 1 Π ] is x n 1 1 x 1 r 1 x n x r y 1 f 1 x 1 x y t f t x 1 x F[y 1 y t ] the polynomial f i x 1 x is the representer of A i i 1 t. Proof Consider the following F - algebra homomorphism φ : F[y 1 y t ] F[x 1 x ]/ x n 1 1 x 1 r 1 x n x r F[Π 1 Π ] y 1 f 1 x 1 x + x n 1 x r A 1 f 1 Π 1 Π y t f t x 1 x + x n 1 x r A t f t Π 1 Π. It is clear that φgy 1 y t 0 if and only if ga 1 A t 0. Hence by Lemma 8 IA 1 A t erφ J F[y 1 y t ]. According to Theorem 13 we give the following algorithm for the annihilation ideal of level FLS r 1 r -circulate matrices A 1 A t F[Π 1 Π ]. E-ISSN: Issue 4 Volume 1 April 013

7 Step 1 Calculate the reduced Gröbner basis G for the ideal x n 1 x r y 1 f 1 x 1 x y t f t x 1 x by CoCoA 4.0 using an elimination order with x 1 > > x > y 1 > > y. Step Find the polynomial in G in which the variables x 1 x x are not appear. Then the ideal generated by these polynomials is the annihilation ideal of A 1 A t. To calculate the common minimal polynomial of A 1 A t we first give the following Lemma. Lemma 14 Let hx be the least common multiple of p 1 x p x p x. Then Proof: For any we have p i x hx. i1 fx p i x i1 p i x fx for i 1. Since hx is the least common multiple of p 1 x p x p x hx fx. So Hence fx hx. p i x hx. i1 Conversely p i x hx for i 1 because hx is the least common multiple of p 1 x p x p x. Therefore p i x hx. i1 By Lemma 14 and Lemma 10 If the minimal polynomial of A i is p i x for i 1 t then the common minimal polynomial of A 1 A t is the least common multiple of p 1 x p x p t x. So we have the following algorithm for the common minimal polynomial of level- FLS r 1 r -circulant matrices A i f i Π 1 Π for i 1 t: Step 1 Calculate the Gröbner basis G i for the ideal x n 1 1 x 1 r 1 x n x r y f i x 1 x by CoCoA 4.0 for each i 1 t using an elimination order with x 1 > > x > y. Step Find out the polynomial g i y in G i in which the variables x i x do not appear for each i 1 t. Step 3 Calculate the Gröbner basis G for the ideal 1 t i1 w i w 1 g 1 y w t g t y by CoCoA 4.0 using elimination with w 1 > > w t > y. Step 4 Find out the polynomial gy in G in which the variables w 1 w t do not appear. Then the polynomial gy is the common minimal polynomial of A i f i Π 1 Π for i 1 t. Example 15 Let A 1 f 1 Π 1 Π and A f Π 1 Π be both level circulant matrices Π 1 ℵ 11 I 4 Π I 4 ℵ 14 f 1 x y x 3 y 3 +x 3 y +x 3 y+3x 3 +7x y 3 +4x y +3x y+x +xy 3 +7xy +xy+6x+y 3 +3y +y+5 f x y x 3 y 3 +x 3 y +3x 3 y+x 3 +6x y 3 +5x y +7x y+x +4xy 3 +3xy +xy+4x+6y 3 +3y +y+4 and ℵ 14 ℵ I We calculate the common minimal polynomial of A 1 and A in the field Z 11 as following: By CoCoa 4.0 we obtain reduced Gröbner basis for the ideal x 4 x 11 y 4 y 14 z f 1 x y is G 1 {z 1 + z z 10 z 9 + 5z 8 4z 7 + 4z 6 +3z 5 + z 4 3z 3 + z + 5z + 1 x x yz 4yz 4y 5z 11 z 10 + z 9 + z 8 + 4z 7 + 4z 6 z 5 + z 4 z 3 + 4z + 5z + xy + 5x + 5yz + y + 3z 11 3z z 9 z 8 5z 7 z 6 + 5z 5 + 4z 4 + z 3 3z + 5z + 5 xz + 4x yz + 3yz y + z 11 + z z 9 5z 8 5z 7 z 6 3z 5 4z 4 3z 3 + 4z + 5z y + yz + 5yz y + 5z 11 + z 10 z 9 5z 8 5z 7 z 6 + z 5 + z 3 + 3z + z 5 yz 3 yz + 3yz 4y + 3z z 10 z 9 5z 7 z 6 5z 5 + 4z 4 + z 3 + 5z + 3}. E-ISSN: Issue 4 Volume 1 April 013

8 So the minimal polynomial p 1 z of A 1 is z 1 + z z 10 z 9 + 5z 8 4z 7 +4z 6 + 3z 5 + z 4 3z 3 + z + 5z + 1. By CoCoa 4.0 we get the reduced Gröbner basis for the ideal x 4 x 11 y 4 y 14 z f x y is G {z 15 z 14 z 13 3z 1 + z 11 z 10 +z 9 z 8 + z 7 + 3z 6 + z 5 + 5z 5z x + x z 14 z 13 z 1 + z 11 5z z 9 +4z 8 z 7 4z 5 + z 4 + 4z 3 + 5z + z xz 3x + z z 13 + z 1 + 5z 11 + z z 9 4z 8 + 5z 7 + z 6 + z 5 + 3z 4 z 3 4z + 5z y + 3z 14 3z 1 + 3z 11 + z 10 3z 9 + 4z 8 3z 7 3z 6 5z 5 z 4 5z 3 z + 3z }. So the minimal polynomial p z of A is z 15 z 14 z 13 3z 1 + z 11 z 10 +z 9 z 8 + z 7 + 3z 6 + z 5 + 5z 5z. By CoCoa 4.0 we obtain the reduced Gröbner basis for the ideal 1 u v up 1 z vp z is G {u + v 1 vz 4 4vz 3 5vz 4vz + 3v 5z 3z 1 + z 0 z z z 17 + z z 15 4z 14 +5z 13 z 1 + z 11 + z 10 z 9 4z 8 z 7 + 3z 6 +4z 5 + 3z 4 5z 3 + 5z + z 3 z 3 + 5z + 4z 1 z 0 5z z 17 z 16 +3z z 14 4z 13 + z 1 + z 11 + z 10 +3z 9 z 8 z 7 5z 6 + z 4 + 3z 3 4z + z}. So the common minimal polynomial pz of A 1 and A is z 3 + 5z + 4z 1 z 0 5z z 17 z 16 +3z z 14 4z 13 + z 1 + z 11 + z 10 +3z 9 z 8 z 7 5z 6 + z 4 + 3z 3 4z + z. 4 Efficient algorithms for finding the inverses of level- FLS r 1 r - circulant matrices In this section we discuss the singularity and the inverse of a level- FLS r 1 r -circulant matrix. Theorem 16 Let A F[Π 1 Π ] be an N N level- FLS r 1 r -circulant matrix. Then A is nonsingular if and only if 1 fx 1 x x n 1 1 x 1 r 1 x n x r the polynomial fx 1 x is the representer of A. Proof. A is nonsingular if and only if fx 1 x + x n 1 x r is an invertible element in Fx 1 x / x n 1 By Theorem 5 if and only if there exists hx 1 x + x n 1 x r F[x 1 x ]/ x n 1 x r hx 1 x fx 1 x + x n 1 x r 1 + x n 1 x r if and only if there exist hx 1 x u 1 u F[x 1 x ] hx 1 x fx 1 x + u 1 x n 1 1 x 1 r u x n x r 1 if and only if 1 fx 1 x x n 1 1 x 1 r 1 x n x r. Let A F[Π 1 Π ] be an N N level- FLS r 1 r -circulant matrix by Theorem 16 we have the following algorithm which can find the inverse of the matrix A: Step 1 Calculate the reduced Gröbner basis G for the ideal fx 1 x x n 1 x r the polynomial fx 1 x is the representer of A by CoCoA 4.0 using a given term order with x 1 > > x. If G {1} then A is singular. Stop. Otherwise go to step. Step By Buchberger s algorithm for computing Gröbner bases eeping trac of linear combinations E-ISSN: Issue 4 Volume 1 April 013

9 that give rise to the new polynomials in the generating set we get hx 1 x u 1 u F[x 1 x ] hx 1 x fx 1 x + u 1 x n 1 1 x 1 r u x n x r 1 4 Step 3 The variables x 1 x in the above formula 4 are replaced by Π 1 Π respectively we have A 1 hπ 1 Π. 5 Inverse of partitioned matrix with level- FLS r 1 r -circulant matrix blocs Let A 1 A A 3 A 4 be level- FLS r 1 r - circulant matrices with the representer f 1 x 1 x f x 1 x f 3 x 1 x f 4 x 1 x respectively. If A 1 is nonsingular let Γ 1 then I 0 A 3 A 1 1 I Γ 1 ΣΓ A1 A Σ A 3 A 4 I A 1 Γ 1 A 0 I A1 0 0 A 4 A 3 A 1 1 A. 5 So Σ is nonsingular if and only if A 4 A 3 A 1 1 A is nonsingular. Since A 1 A A 3 A 4 are all level- FLS r 1 r -circulant matrices then the A i commutes with the A f i j. Thus A 1 A 4 A 3 A 1 1 A A 1 A 4 A A 3. 6 By the equation 6 we conclude that Σ is nonsingular if and only if A 1 A 4 A A 3 is nonsingular. Since f 1 x 1 x f 4 x 1 x f x 1 x f 3 x 1 x is the representer of A 1 A 4 A A 3 then Σ is nonsingular if and only if 1 f 1 x 1 x f 4 x 1 x f x 1 x f 3 x 1 x x n 1 In addition if Σ is nonsingular by the equation 5 we have I A Σ 1 A 0 I A I 0 0 A 4 A 3 A 1 1 A A 3 A 1 1 I T1 T T 3 T 4 T 1 A A 1 A 4 A A 3 1 A A 3 A 1 T A 1 A 4 A A 3 1 A T 3 A 1 A 4 A A 3 1 A 3 T 4 A 1 A 4 A A 3 1 A 1. Therefore we have the following result. Theorem 17 Let A1 A Σ A 3 A 4 1 A 1 A A 3 and A 4 are all level- FLS r 1 r -circulant matrices with the representer f 1 x 1 x f x 1 x f 3 x 1 x f 4 x 1 x respectively. If A 1 is nonsingular then Σ is nonsingular if and only if 1 f 1 x 1 x f 4 x 1 x f x 1 x f 3 x 1 x x n 1 Moreover if Σ is nonsingular then Σ 1 T1 T. 7 T 3 T 4 T 1 A A 1 A 4 A A 3 1 A A 3 A 1 T A 1 A 4 A A 3 1 A T 3 A 1 A 4 A A 3 1 A 3 T 4 A 1 A 4 A A 3 1 A 1. Theorem 18 Let A1 A Σ A 3 A 4 1 A 1 A A 3 A 4 are all level- FLS r 1 r -circulant matrices with the representer f 1 x 1 x f x 1 x f 3 x 1 x f 4 x 1 x respectively. If A 4 is nonsingular then Σ is nonsingular if and only if 1 f 1 x 1 x f 4 x 1 x f x 1 x f 3 x 1 x x n 1 E-ISSN: Issue 4 Volume 1 April 013

10 Moreover if Σ is nonsingular then Σ 1 T1 T 8 T 3 T 4 T 1 A 1 A 4 A A 3 1 A 4 T A 1 A 4 A A 3 1 A T 3 A 1 A 4 A A 3 1 A 3 T 4 A A 1 A 4 A A 3 1 A A 3 A 1 4. Proof. Since A 4 is nonsingular then I A A 1 4 I 0 Σ 0 I A 1 4 A 3 I A1 A A 1 4 A A 4. 9 So Σ is nonsingular if and only if A 1 A A 1 4 A 3 is nonsingular. Since A 1 A A 3 A 4 are all level- FLS r 1 r -circulant matrices then the A i commutes with the A f i j. Thus A 4 A 1 A A 1 4 A 3 A 1 A 4 A A By the equation 10 we conclude that Σ is nonsingular if and only if A 1 A 4 A A 3 is nonsingular. Since f 1 x 1 x f 4 x 1 x f x 1 x f 3 x 1 x is the representer of A 1 A 4 A A 3 then Σ is nonsingular if and only if 1 f 1 x 1 x f 4 x 1 x f x 1 x f 3 x 1 x x n 1 In addition if Σ is nonsingular by the equation 9 we have Σ 1 I 0 A 1 4 A 3 I A1 A A 1 4 A T1 T T 3 T 4 0 A 1 4 I A A I T 1 A 1 A 4 A A 3 1 A 4 T A 1 A 4 A A 3 1 A T 3 A 1 A 4 A A 3 1 A 3 T 4 A A 1 A 4 A A 3 1 A A 3 A 1 4. We have the following algorithm for determining the nonsingularity and computing the inverse of Σ if it is nonsingular. Step 1 Calculate the Gröbner bases G 1 G 4 for the ideals f 1 x 1 x x n 1 x r f 4 x 1 x x n 1 x r respectively. If G 1 {1} G 4 {1} Stop. Otherwise go to step. Step. If G 1 {1} find u 1 u h 1 x 1 x F[x 1 x ] h 1 x 1 x f 1 x 1 x + u 1 x n 1 1 x 1 r u x n x r 1. Then h 1 x 1 x is the representer of A 1 1 go to step 4. Otherwise go to step 3. Step 3. If G 4 {1} find u 1 u h 4 x 1 x F[x 1 x ] h 4 x 1 x f 4 x 1 x + u 1x n 1 1 x 1 r u xn x r 1. Then h 4 x 1 x is the representer of A 1 4 go to step 4. Step 4. Calculate the Gröbner bases G for the ideals f 1 x 1 x f 4 x 1 x f x 1 x f 3 x 1 x x n 1 If G {1} then A 1 A 4 A A 3 is singular Stop. Otherwise go to step 5. Step 5. Find v 1 v hx 1 x F[x 1 x ] hx 1 x [f 1 x 1 x f 4 x 1 x f x 1 x f 3 x 1 x ] +v 1 x n 1 1 x 1 r v x n x r 1. Then hx 1 x is the representer of A 1 A 4 A A 3 1. Then we obtain If A 1 is nonsingular then Σ 1 T1 T ; T 3 T 4 if A 4 is nonsingular then Σ 1 T5 T T 3 T 6 E-ISSN: Issue 4 Volume 1 April 013

11 T 1 h 1 Π 1 Π [I + hπ 1 Π A A 3 ] T hπ 1 Π A T 3 hπ 1 Π A 3 T 4 hπ 1 Π A 1 T 5 hπ 1 Π A 4 T 6 h 4 Π 1 Π [I + hπ 1 Π A A 3 ]. Acnowledgements:The project is supported by the Development Project of Science & Technology of Shandong Province Grant Nos. 01GGX References: [1] S. S. Capizzano and E. Tyrtyshniov Any circulant-lie perconditioner for multilevel matrices is not superlinear SIAM J. Matrix. Anal. Appl pp [] S. S. Capizzano and E. Tyrtyshniov Multigrid methods for multilevel circulant matrices SIAM J. Sci. Comput pp [3] R. M. Mallery Reverse Engineering Gene Networs with Microarray Data. cox/genenets.pdf. [4] G. Songa and Y. Xua Approximation of highdimensional ernel matrices by multilevel circulant matrices Journal of Complexity pp [5] W. F. Trench Properties of unilevel bloc circulants Linear Algebra and its Appl pp [6] W. F. Trench Properties of multilevel bloc α- circulants Linear Algebra and its Appl pp [7] S. G. Zhang Z. L. Jiang and S. Y. Liu An application of the Gröbner basis in computation for the minimal polynomials and inverses of bloc circulant matrices Linear Algebra and its Appl pp [8] M. Morhac and V. Matouse Exact algorithm of multidimensional circulant deconvolution Appl. Math. and Comput pp [9] M. Rezghi and L. Elden Diagonalization of tensors with circulant structure Linear Algebra and its Appl pp [10] S. Georgiou and C. Kououvinos Multi-level -circulant supersaturated designs Metria pp [11] Z. L. Jiang and S. Y. Liu Level- m scaled circulant factor matrices over the complex number field and the quaternion division algebra J. Appl. Math. Computing pp [1] A. J. H Bloc circulants of level-. Division of applied Mathematics. Brown University providence R. I [13] J. Baer The structure of multi-bloc circulants Kyungpoo Math. J pp [14] R. L. Smith Moore-Penrose inverses of bloc circulant and bloc -circulant matrices Linear Algebra and its Appl pp [15] S. Rjasanow Effective algorithms with circulant-bloc matrices Linear Algebra and its Appl pp [16] W. W. Adams and P. Loustaunau An introduction to Gröbner bases American Mathematical Society [17] B. Mishra Algorithmic Algebra Springer- Verlag 001. [18] D. G. Northcott Injective envelopes and inverse polynomials J. London Math. Soc pp [19] D. Greenspan Methods of matrix inversion Amer. Math. Monthly pp [0] H. M. and T. Mora New constructive methods in classical ideal theory J. Algebra pp [1] P. J. Davis Circulant Matrices Wiley New Yor [] R. A. Horn and C. R. Johnson Matrix Analysis New Yor: Cambridge University Press [3] I. Gohberg P. Lancaster and L. Rodman Matrix Polynomials Academic Press New Yor 198. [4] Z. L. Jiang Z. X. Zhou Circulant Matrices Chengdu Technology University Publishing Company Chengdu [5] Z. L. Jiang Z. B. Xu Eifficient algorithm for finding the inverse and group inverse of FLS r- circulant matrix J. Appl. Math. Comput pp E-ISSN: Issue 4 Volume 1 April 013