A closed-form expression for the kth power of semicirculant and r-circulant matrices
aa r X i v : . [ m a t h . R A ] J un A CLOSED-FORM EXPRESSION FOR THE KTHPOWER OF SEMICIRCULANT AND r -CIRCULANTMATRICES M. MOUC¸ OUFDepartment of Mathematics, Faculty of Science, Chouaib Doukkali University,MoroccoEmail: [email protected]
Abstract.
We derive a closed-form expression for the kth powerof semicirculant matrices by using the determinant of certain ma-trices. As an application, a closed-form expression for the kthpower of r -circulant matrices is also povided. Introduction
The n × n r -circulant matrix C n,r over a unitary commutative ring R is one having the following form(1.1) C n,r = ⎛⎜⎜⎜⎜⎜⎝ c c c ⋯ c n − c n − rc n − c c ⋯ c n − c n − rc n − rc n − c ⋯ c n − c n − ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ rc rc rc ⋯ rc n − c ⎞⎟⎟⎟⎟⎟⎠ , where r ∈ R is a parameter. The r -circulant matrix C n,r is deter-mined by r and its first row elements c , . . . , c n − , thus we denote C n,r = circ n,r ( c , . . . , c n − ) .However, an infinite semicirculant matrix over R is one having thefollowing form (see e.g., Henrici [3] or Davis [2]) A = [ a , a , a , . . . ] = ⎛⎜⎜⎜⎜⎜⎝ a a a a ⋯ a a a ⋯ a a ⋯ a ⋯⋯ ⋯ ⋯ ⋯ ⋯ ⎞⎟⎟⎟⎟⎟⎠ . Mathematics Subject Classification.
Key words and phrases.
Semicirculant, r -Circulant, Powers, Sequence,Determinant. In the work [1], the general expression of the k th power A k = [ a ( k ) , a ( k ) , a ( k ) , . . . ] ( k ∈ N ) of a semicirculant matrix A = [ a , a , a , . . . ] is presented.More precisely the sequence { a m ( k )} m ≥ is obtained using a recursivemethod. To do this, we have proved that for all k ∈ N ,(1.2) a i ( k ) = L ( A )( i, ) a k ( k ) + ⋯ + L ( A )( i, j ) a k − j ( kj ) + ⋯ + L ( A )( i, i ) a k − i ( ki ) , where we adopt the convention that for any element a ∈ R and anynonnegative integers k ≤ p , a k − p ( kp ) = δ k,p .The double sequence { L ( A )( n, m )} n,m of elements of R is defined by L ( A )( n, m ) = ∑ ∆ ( n,m ) ( mk , . . . , k n ) a k ⋯ a k n n , (1.3)where ∆ ( n, m ) , m ⩽ n are integers, is the solution set of the followingsystem of equations ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩( k , . . . , k n ) ∈ N n k + ⋯ + k n = mk + k + ⋯ + nk n = n. We note that Formula (1.2) expresses a i ( k ) in closed-form that is noteasy to use. Here we provide an easy closed-form formula in terms ofdeterminants. As a consequence, in the last section we give a closed-form formula for the kth power of r -circulant matrices and we describea method for finding the solution set of the well known Diophantineequation k + k + ⋯ + nk n = n. Throughout this paper R will denote an arbitrary commutative ringwith identity.2. Sequences L ( A )( n, m ) and u n ( A ) corresponding to asemicirculant matrix A Let A = [ a , a , a , . . . ] be a semicirculant matrix over R . Then Thesequence ( L ( A )( i, j )) i,j is independent on the coefficient a and it isuniquely determined by the following recursive formula (see Lemma 2.2of [1]):(2.1) ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ L ( A )( , ) = L ( A )( i, ) = i ≠ L ( A )( i, j ) = i < jL ( A )( i, j + ) = a L ( A )( i − , j ) + ⋯ + a i − j L ( A )( j, j ) OWERS OF SEMICIRCULANT AND r -CIRCULANT MATRICES 3 Let L ( A ) be the matrix ( L ( A )( i, j )) ⩽ i,j . Then L ( A ) is the followinglower triangular matrix10 a a a a a a a a a a + a a a a a a a + a a a a + a a a a a a a a + a a + a a a + a a a + a a a + a a a a a ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ Each colomn of this matrix can be deduced from the precedent one.For example ⟨( a , a , a , a ) , ( a a + a a , a a + a , a a , a )⟩ = a a + a a a + a . Let L i ( A ) be the i th row of the matrix L ( A ) and let S i ( A ) be the rightshift of L i ( A ) by one position. Then formula (2.1) implies that(2.2) S n ( A ) = a L n − ( A ) + ⋯ + a n L ( A ) . We now consider the sequence(2.3) u n ( A ) = { ∑ ni = L ( A )( n, i ) u ( A ) = L ( A )( , ) = u n is the sum of all elements of the ( n + L ( A ) . From formula (2.2) it follows that(2.4) { u n ( A ) = a u n − ( A ) + a u n − ( A ) + ⋯ + a n u ( A ) u ( A ) = u n ( A ) is uniquely determined by the relation(2.4).Let us now consider the following sets: S C the set of all infinite semicirculant matrices with entries in the ring R , D the subset of S C consisting of all diagonal matrices, U the set of all sequences satisfying the recurrence relation (2.4) forsome sequences { a n } n ≥ of the elements of R , V the set of all double sequences satisfying the recurrence relation (2.1)for some sequences { a n } n ≥ of the elements of R . M. MOUC¸ OUF
The following result shows the existence of a biunivoque correspon-dence between any two of the sets: U , V and S C/D . Proposition 2.1.
There is a commutative diagram of bijections
S C/DV U ̺ψ ϕ where the maps ρ , ψ and ϕ are defined as follows ̺ ( A ) = { u n ( A )} n ≥ , A denotes the equivalence class of A ∈ S C modulo D .ψ ( A ) = { L ( A )( n, m )} n,m ,ϕ ({ φ ( n, m )} n,m ) = { n ∑ i = φ ( n, i )} n ≥ . These maps have respectively as inverses the following ψ − ({ φ ( n, m )} n,m ) = [ , φ ( , ) , . . . , φ ( n, ) , . . . ] ,̺ − ({ u n } n ≥ ) = [ , a , . . . , a n , . . . ] ,ϕ − ({ u n } n ≥ ) = { φ ( n, m )} n,m , where φ ( n, m ) = ∑ ∆ ( n,m ) ( mk , . . . , k n ) a k ⋯ a k n n and { a n } n ≥ is the sequenceof elements of R corresponding to { u n } n ≥ .Proof. Because sequences of U are uniquely determined by (2.4), themap ̺ is well defined. Let now A = [ a , a , a , . . . ] and B = [ b , b , b , . . . ] such that u n ( A ) = u n ( B ) , for all positive integer n . Then u ( A ) = u ( B ) and hence a = b . Furthermore, the relation (2.4) implies that a n = a n u ( A ) = u n ( A ) − a u n − ( A ) − ⋯ − a n − u ( A ) . It then follows byan easy induction that a n = b n for all positive integer n , i.e., A = B .Hence the map ̺ is injective. To show that ̺ is surjective, consider anelement ( u n ) n ∈ N of U and let ( a n ) n ∈ N ∗ the corresponding sequence of R .It is clear that ̺ ([ , a , . . . , a n , . . . ]) = { u n } n ≥ , and then the map ̺ isbijective.The same argument as for ̺ applies again to ψ .Now let { φ ( n, m )} n,m ∈ V and consider the matrix A = [ , φ ( , ) , . . . , φ ( n, ) , . . . ] .It is clear that ϕ ({ φ ( n, m )} n,m ) = { u n ( A )} n ≥ and then ϕ is bijective.The last assertion follows immediately from the fact that ϕ ○ ψ = ρ . ∎ OWERS OF SEMICIRCULANT AND r -CIRCULANT MATRICES 5 Proposition 2.2.
Let a , a , a , . . . ∈ R and let δ ( a , a , . . . , a n ) be thefollowing determinant (2.5) δ ( a , a , . . . , a n ) = RRRRRRRRRRRRRRRRRRRRRRRRRRR a a ⋯ ⋯ ⋯ a n a ⋱ ⋱ ⋮ ⋱ ⋱ ⋱ ⋮⋮ ⋱ ⋱ ⋱ ⋱ ⋮⋮ ⋱ ⋱ ⋱ a ⋯ ⋯ a a RRRRRRRRRRRRRRRRRRRRRRRRRRR . Then we have δ ( a , a , . . . , a n ) = δ ( , a , . . . , a i a i − , . . . , a n a n − ) (2.6) = δ ( − , a , . . . , a i ( − a ) i − , . . . , a n ( − a ) n − ) (2.7) for all n ∈ N ∗ .Proof. The claim is trivially true for a =
0. Suppose a ≠
0. For theproof of the first equality it suffices to multiply every column C i of thedeterminant δ ( a , a , . . . , a n ) by a i − , and then multiply every row R i of the resulting determinant by a − i + .To obtain the last equality, it suffices to multiply each of the even rowsof the determinant δ ( , . . . , a i a i − , . . . , a n a n − ) by −
1, and then multiplyeach of the even columns of the resulting determinant by − ∎ Proposition 2.3.
Let A = [ a , a , a , . . . ] be a semicirculant matrixover R , and let { u n ( A )} n ≥ be the sequence of elements of R associatedto A given in (2.3) . Then we have (2.8) u n ( A ) = δ ( − , a , . . . , a n ) for all n ∈ N ∗ .Proof. By (2.6) we know that δ ( − , a , . . . , a n ) = δ ( , a , . . . , a i ( − ) i − , . . . , a n ( − ) n − ) .Assume that n ⩾
1, and let us expand the determinant δ ( , a , . . . , a i ( − ) i − , . . . , a n ( − ) n − ) along the first row. Then we get δ ( − , a , . . . , a n ) = n − ∑ i = a i δ i , where δ i is the cofactor associated with the entry ( − ) i − a i of the matrix δ ( , a , . . . , a i ( − ) i − , . . . , a n ( − ) n − ) . M. MOUC¸ OUF
It can be easily seen that the cofactor δ i has the form δ i = RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR ⋆ ⋆ ⋱ a − a ⋯ a n − i ( − ) n + − i ⋱ ⋮⋱ ⋱ ⋮ a RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR
Thus for i ⩾ δ i = δ ( − , a , . . . , a n − i − ) . Hence δ ( − , a , . . . , a n ) = n − ∑ i = a i δ ( − , a , . . . , a n − i − ) . It follows that the sequence δ ( − , a , . . . , a n ) satisfies the recurrencerelation (2.4). But since δ ( − , a ) = a , we have δ ( − , a , . . . , a n ) = u n for all n ∈ N ∗ . ∎ The following proposition is useful for the proof of the main result.
Proposition 2.4.
Let X be an indeterminate over R and let { a n } n ≥ bea sequence of the elements of R . Let X n ( X ) be the polynomial sequencedefined by (2.9) X n ( X ) = δ ( − X, a , . . . , a n ) . Then one has (2.10) X n ( X ) = n ∑ i = L ( A )( n, i ) X n − i , where A = [ , a , a , . . . ] .Proof. Let A = [ , a , a , . . . ] and consider the semicirculant matrix B = [ , a , a X, . . . , a n X n − , . . . ] over R [ X ] . Then L ( B )( n, m ) = ∑ ∆ ( n,m ) ( mk , . . . , k n ) a k ( a X ) k ⋯ ( a n X n − ) k n = ∑ ∆ ( n,m ) ( mk , . . . , k n ) a k ⋯ a k n n X k +⋯+( n − ) k n , and since k +⋯+ ( n − ) k n = ( k + k +⋯+ nk n ) − ( k + k +⋯+ k n ) it followsthat k + ⋯ + ( n − ) k n = n − m , and L ( B )( n, m ) = L ( A )( n, m ) X n − m .On the other hand, Formula (2.7) together with Formula (2.8) yield u n ( B ) = X n ( X ) . As a consequence of Proposition 2.1, we have X n ( X ) = n ∑ i = L ( B )( n, i ) = n ∑ i = L ( A )( n, i ) X n − i . ∎ OWERS OF SEMICIRCULANT AND r -CIRCULANT MATRICES 7 Let k and n be nonnegative integers and let [ . ] nk ∶ R [ X ] Ð → R [ X ] be the linear map defined by [ X i ] nk = X k −( n − i ) ( kn − i ) , where we use the convention that for k ≤ p ,(2.11) X k − p ( kp ) = δ k,p , the Kronecker delta. Then we have The following result which gives aclosed-form expression of the kth power of semicirculant matrices. Theorem 2.5.
Let { a n } n ∈ N be a sequence of the elements of R . Forall nonnegative integer k , the kth power of the semicirculant matrix A = [ a , a , a , . . . ] are given as follows: A k = [[X ] k ( a ) , [X ] k ( a ) , [X ] k ( a ) , . . . ] (2.12) where X n is the polynomial δ ( − X, a , . . . , a n ) given in (2.9) and X = .Proof. Follows immediately from Proposition 2.4 and Formula (1.2). ∎ Example 2.6.
Consider a semicirculant matrix A = [ , , , − , ] . Let k be any nonnegative integer. We have X ( X ) = RRRRRRRRRRRRR − − X − X RRRRRRRRRRRRR = − X + X + . Then the ( , ) entry of matrix A k is − × k − ( k ) + × k − ( k ) + × k − ( k ) . Example 2.7.
Consider another semicirculant matrix B = [ , , , , ] .Let k be any nonnegative integer. We have X ( X ) = RRRRRRRRRRRRRRRRRR − X − X − X RRRRRRRRRRRRRRRRRR = x + x + x + . Then the ( , ) entry of matrix B k is ( k ) k − + ( k ) k − + ( k ) k − + ( k ) k − . Applications
A closed-form expression for the kth power of C n,r . M. MOUC¸ OUF
Theorem 3.1.
Let r, c , . . . , c n − ∈ R and let k be any nonnegativeinteger. Consider the r -circulant matrix C n,r = circ n,r ( c , . . . , c n − ) . Then the pth strip of C kn,r is ( C kn,r ) p = ∑ m ≡ p ( mod n ) c k − m [X m ] mk ( c ) r ⌊ mn ⌋ , where X m ( X ) = { δ ( − X, c , . . . , c m ) if m ≥ if m = , and ⌊ x ⌋ denotes the greatest integer less than or equal to x .Proof. Follows immediately from Theorem 2.5 and Theorem 4.1 of [1]. ∎ Remark 3.2.
The sequence r ⌊ mn ⌋ , appearing in the pth strip ( C kn,r ) p = ∑ m ≡ p c m ( k ) r ⌊ mn ⌋ of the r -circulant matrix C kn,r , is nothing but the geometric sequencewith common ratio r . Example 3.3.
Let C = circ , ( , , , , ) and let [ C ] = [ , , , , , , , . . . ] be the associated infinite semicirculant matrix. Put [ C ] k = [ c ( k ) , c ( k ) , c ( k ) , . . . ] . OWERS OF SEMICIRCULANT AND r -CIRCULANT MATRICES 9 Using the method provided in Theorem 2.5, we obtain c s ( ) = for all s ≥ ( × ) + = X ( X ) = x + ⋯ , c ( ) = × − ( ) = X ( X ) = x + ⋯ , c ( ) = × ( ) = X ( X ) = X + ⋯ , c ( ) = × ( ) = X ( X ) = X + ⋯ , c ( ) = × ( ) = X ( X ) = x + x + ⋯ , c ( ) = ( ) + × ( ) = X ( X ) = X + X + ⋯ , c ( ) = × ( ) + × ( ) = X ( X ) = X + X + ⋯ , c ( ) = × ( ) + × ( ) = X ( X ) = X + X + ⋯ , c ( ) = × ( ) + × ( ) = X ( X ) = x + x + x + ⋯ , c ( ) = ( ) + × ( ) + × ( ) = X ( X ) = X + X + , c ( ) = × ( ) + × ( ) + × ( ) = X ( X ) = X + , c ( ) = × ( ) + × ( )) = X ( X ) = , c ( ) = × ( ) = c ( ) = = . Hence [ C ] = [ , , , , , , , , , , , , , , , . . . ] .Therefore, C = × + × + × = C = × + × + × = C = × + × + = C = × + × = C = × + × = . Thus C = circ , ( , , , , ) . The calculation of ∆ ( n ) . Consider the solution set ∆ ( n, p ) ofthe following system of equations ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩( k , . . . , k n ) ∈ N n k + ⋯ + k n = pk + k + ⋯ + nk n = n. Here, p and n are positive integers such that p ⩽ n .Let ∆ ( n ) be the solution set of the diophantine equation k + k + ⋯ + nk n = n. We have clearly ∆ ( n ) = ⋃ ≤ i ≤ n ∆ ( n, i ) . Let
X, x , . . . , x n be independent indeterminates over R . From (2.10)we have X n ( X ) = n ∑ i = L ( A )( n, i ) X n − i , and from (1.3) we have L ( A )( n, i ) = ∑ ∆ ( n,i ) ( ik , . . . , k n ) x k ⋯ x k n n , where A = [ , x , . . . , x n ] .We observe that we can determine the set ∆ ( n ) by computing thepolynomial X n ( X ) , and then Comparing the polynomial L ( A )( n, i ) , ≤ i ≤ n, and the term of degree n − i of X n ( X ) . Example 3.4.
We have X ( X ) = x X + ( x x + x ) X + x x X + x .Then ∆ ( , ) = {( , , , )} , ∆ ( , ) = {( , , , ) , ( , , , )} , ∆ ( , ) = {( , , , )} , ∆ ( , ) = {( , , , )} OWERS OF SEMICIRCULANT AND r -CIRCULANT MATRICES 11 Example 3.5.
We have X ( X ) = x X + ( x x + x x + x ) X + ( x x + x x x + x ) X + ( x x + x x ) X + x x X + x . Then ∆ ( , ) = {( , , , , , )} , ∆ ( , ) = {( , , , , , ) , ( , , , , , ) , ( , , , , , )} , ∆ ( , ) = {( , , , , , ) , ( , , , , , ) , ( , , , , , )} , ∆ ( , ) = {( , , , , , ) , ( , , , , , )} , ∆ ( , ) = {( , , , , , )} , ∆ ( , ) = {( , , , , , )} References [1] M. Mou¸couf,
Arbitrary positive power of semicirculant and r -circulant matri-ces, arXiv:2006.15048 [math.RA].[2] P. Davis, Circulant matrices,
Ams Chelsea Publishing, Providence, Rhode Is-land, 2012.[3] P. Henrici,