Some (Hopf) algebraic properties of circulant matrices
We study some (Hopf) algebraic properties of circulant matrices, inspired by the fact that the algebra of circulant n × n matrices is isomorphic to the group algebra of the cyclic group with n elements. We introduce also a class of matrices that generalize both circulant and skew circulant matri...
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irk-123456789-1521612019-06-09T01:24:49Z Some (Hopf) algebraic properties of circulant matrices Albuquerque, H. Panaite, F. We study some (Hopf) algebraic properties of circulant matrices, inspired by the fact that the algebra of circulant n × n matrices is isomorphic to the group algebra of the cyclic group with n elements. We introduce also a class of matrices that generalize both circulant and skew circulant matrices, and for which the eigenvalues and eigenvectors can be read directly from their entries. 2012 Article Some (Hopf) algebraic properties of circulant matrices / H. Albuquerque, F. Panaite // Algebra and Discrete Mathematics. — 2012. — Vol. 13, № 1. — С. 1–17. — Бібліогр.: 5 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 15B05; 16W30. http://dspace.nbuv.gov.ua/handle/123456789/152161 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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We study some (Hopf) algebraic properties of
circulant matrices, inspired by the fact that the algebra of circulant
n × n matrices is isomorphic to the group algebra of the cyclic
group with n elements. We introduce also a class of matrices that
generalize both circulant and skew circulant matrices, and for which
the eigenvalues and eigenvectors can be read directly from their
entries. |
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Article |
| author |
Albuquerque, H. Panaite, F. |
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Albuquerque, H. Panaite, F. Some (Hopf) algebraic properties of circulant matrices Algebra and Discrete Mathematics |
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Albuquerque, H. Panaite, F. |
| author_sort |
Albuquerque, H. |
| title |
Some (Hopf) algebraic properties of circulant matrices |
| title_short |
Some (Hopf) algebraic properties of circulant matrices |
| title_full |
Some (Hopf) algebraic properties of circulant matrices |
| title_fullStr |
Some (Hopf) algebraic properties of circulant matrices |
| title_full_unstemmed |
Some (Hopf) algebraic properties of circulant matrices |
| title_sort |
some (hopf) algebraic properties of circulant matrices |
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Інститут прикладної математики і механіки НАН України |
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2012 |
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Some (Hopf) algebraic properties of circulant matrices / H. Albuquerque, F. Panaite // Algebra and Discrete Mathematics. — 2012. — Vol. 13, № 1. — С. 1–17. — Бібліогр.: 5 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT albuquerqueh somehopfalgebraicpropertiesofcirculantmatrices AT panaitef somehopfalgebraicpropertiesofcirculantmatrices |
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2025-07-13T02:25:10Z |
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2025-07-13T02:25:10Z |
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1837496814517878784 |
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h.Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 13 (2012). Number 1. pp. 1 – 17
c© Journal “Algebra and Discrete Mathematics”
Some (Hopf) algebraic properties
of circulant matrices
Helena Albuquerque1 and Florin Panaite2
Communicated by I. P. Shestakov
Abstract. We study some (Hopf) algebraic properties of
circulant matrices, inspired by the fact that the algebra of circulant
n × n matrices is isomorphic to the group algebra of the cyclic
group with n elements. We introduce also a class of matrices that
generalize both circulant and skew circulant matrices, and for which
the eigenvalues and eigenvectors can be read directly from their
entries.
Introduction
The starting point of this paper was a result in [1], that arose during
the study of certain multiplicatively closed lattices and so called Brandt
algebras in (twisted) group rings of cyclic groups: it asserts that for a
twisted group ring RFZ3 (where F is an arbitrary map, not necessarily
a two-cocycle) there exist three forms q1, q2, q3, concretely determined,
such that any element x ∈ RFZ3 satisfies the polynomial equation x3 −
q1(x)x
2+ q2(x)x− q3(x) = 0. Moreover, q3(x) is given by the determinant
of a 3× 3 matrix (which, if F is trivial, is a circulant matrix) and, if F is
a two-cocycle, then q1 and q2 are related in a certain (specific) way.
1The first author was partially supported by the Centre for Mathematics of the
University of Coimbra (CMUC)
2The second author was partially supported by the CNCSIS project “Hopf algebras,
cyclic homology and monoidal categories”, contract nr. 560/2009, CNCSIS code ID−69.
2000 Mathematics Subject Classification: 15B05; 16W30.
Key words and phrases: Hopf algebras; (generalized) circulant matrices; Brandt
algebras.
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2 Some algebraic properties of circulant matrices
We wanted to see to what extent this kind of results may be gener-
alized from n = 3 to arbitrary n. It turns out that, even if analogous
forms q1, . . . , qn may exist, not much could be said about them. Thus,
we restricted our study to the case when the map F is trivial (that is,
to ordinary group rings) and, slightly more general, to the case when
F is a two-cocycle that is trivial in the second cohomology group. We
were thus led to consider circulant matrices as well as a certain class of
generalized circulants. By using circulants, we were able to prove that
forms q1, . . . , qn exist on the group ring CZn and have some properties
that generalize the case n = 3. This is done by using the well known
algebra isomorphism between the algebra of n × n circulants and the
group algebra of the cyclic group Zn. We found useful to give this result
a Hopf algebraic interpretation, obtaining along the way a result stating
that the algebra of n× n matrices "factorizes" (in a certain sense) as the
"product" between the algebra of n×n circulants and the algebra of n×n
diagonal matrices. Since the group ring is a Hopf algebra, the algebra of
n× n circulants becomes also a Hopf algebra. We have written down its
Hopf structure and we found that the antipode looks particularly nice: the
antipode applied to a circulant matrix is simply the transpose of it. The
comultiplication ∆ does not look too good, but we could prove however
that if C is a circulant n× n matrix then ∆(C), regarded as an n2 × n2
matrix, is block circulant with circulant blocks. We present also a certain
result (and a concrete example) concerning some lattices in the algebra of
circulant matrices.
In the last section of the paper we introduce a class of generalized
circulants, as follows. Denote the elements of Zn by e1 = 1, e2, . . . , en;
for a given map µ : Zn → C∗, with µ(e1) = 1, denote µ(ei) = µi for all
i ∈ {2, . . . , n}. For c1, . . . , cn ∈ C, denote by circ(c1, . . . , cn;µ2, . . . , µn)
the n×n matrix with c1, c2, . . . , cn in the first row, c1 on the main diagonal
and entry cj−i+1
µiµj−i+1
µj
in any other position (i, j) (we put µ1 = 1).
Certainly, circ(c1, . . . , cn) = circ(c1, . . . , cn; 1, . . . , 1), so the matrices of
the type circ(c1, . . . , cn;µ2, . . . , µn) generalize circulant matrices.They
form an algebra, denoted by Cn
C(µ), which is isomorphic to the twisted
group algebra with two-cocycle induced by µ (this is actually how we
arrived at these matrices). From general results we know that this twisted
group ring is isomorphic to the ordinary group ring, so Cn
C(µ) is isomorphic
to the algebra of n× n circulants. But these matrices generalize not only
circulants, but also skew circulant matrices: indeed, a skew circulant matrix
scirc(c1, . . . , cn) turns out to be circ(c1, . . . , cn;σ, σ
2, . . . , σn−1), where
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H. Albuquerque, F. Panaite 3
σ = cos(π
n
) + isin(π
n
). Moreover, for a matrix circ(c1, . . . , cn;µ2, . . . , µn)
we are able to give an explicit formula for its eigenvalues (and eigenvectors)
that can be read directly from its entries; this formula is a common
generalization for the formulae (cf. [3]) giving the eigenvalues for circulant
and skew circulant matrices.
1. Preliminaries
We work over a base field K of characteristic zero (so all matrices have
entries in K and all algebras and Hopf algebras are over K). Sometimes
we will need to choose K to be the field C of complex numbers, and we
will specify this every time we do it.
When we work with n× n circulant matrices or with elements in the
cyclic group of order n, all subscripts will be considered mod n. For a
matrix A we denote by AT its transpose.
A circulant matrix of order n is an n × n matrix with the property
that each row is obtained from the previous row by rotating once to the
right. Obviously such a matrix is completely determined by its first row.
If the first row is (c1, c2, . . . , cn) we will denote the associated circulant
matrix by circ(c1, c2, . . . , cn). Note that the (i, j) entry of this matrix is
cj−i+1. For instance, for n = 3, the matrix circ(c1, c2, c3) is
c1 c2 c3
c3 c1 c2
c2 c3 c1
If we denote by Cn
K the set of circulant n× n matrices, then Cn
K is an
n dimensional subalgebra of the algebra Mn(K) of n× n matrices over K.
If K = C, the field of complex numbers, and C = circ(c1, c2, . . . , cn)
is a complex circulant matrix, then the eigenvalues and eigenvectors of
C may be written down explicitely (cf. [3]): if we denote by pC(X) =
c1+ c2X+ . . .+ cnX
n−1 and ω = cos(2π
n
)+ isin(2π
n
), then the eigenvalues
of C are the scalars λj = pC(ω
j−1), for 1 ≤ j ≤ n, the eigenvector of λj
being the vector xj = (1, ωj−1, ω2(j−1), . . . , ω(n−1)(j−1))T .
For more properties of circulant matrices we refer to [3], while for
terminology, notation etc. concerning Hopf algebras we refer to [4].
2. Forms associated to circulant matrices
We consider the group Zn (cyclic group of order n) and we denote its
elements by ei = î− 1, for i ∈ {1, 2, . . . , n}, so we have eiej = ei+j−1 and
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4 Some algebraic properties of circulant matrices
e−1
i = en−i+2, for all i, j ∈ {1, 2, . . . , n} (subscripts mod n, according to
our convention).
We consider the group algebra KZn over the base field K. The following
result is well known:
Theorem 2.1. The group algebra KZn and the algebra Cn
K of circulant
n × n matrices over K are isomorphic, an explicit isomorphism being
defined by g : KZn → Cn
K, g(c1e1 + . . .+ cnen) = circ(c1, c2, . . . , cn), for
all c1, . . . , cn ∈ K.
Corollary 2.2. Every element x = c1e1 + . . .+ cnen ∈ KZn is a solution
to a certain polynomial Xn − q1(x)X
n−1 + . . . + (−1)nqn(x). Here q1
is a linear form called the trace of x and qn is an n-form called the
norm of x. Moreover, q1(x) = tr[circ(c1, . . . , cn)] = nc1 and qn(x) =
det[circ(c1, . . . , cn)].
Proof. The polynomial is exactly the characteristic polynomial of the
circulant matrix corresponding to x via the isomorphism g : KZn ≃
Cn
K.
Remark 2.3. Via the isomorphism KZn ≃ Cn
K, we can also consider the
forms q1, . . . , qn as being defined on Cn
K.
Definition 2.4. For x ∈ KZn, we define the conjugate of x by the formula
x = (−1)n+1xn−1 + (−1)nq1(x)x
n−2 + . . .+ qn−1(x) ∈ KZn. (2.1)
As a consequence of Theorem 2.1 and Corollary 2.2, we immediately
obtain:
Corollary 2.5. An element x ∈ KZn is invertible if and only if qn(x) 6= 0,
and in this case its inverse is defined by the formula x−1 = x
qn(x)
.
Corollary 2.6. An element x = c1e1 + . . .+ cnen ∈ CZn is noninvertible
if and only if there exists y a root of unity of order n such that c1 + c2y +
. . .+ cny
n−1 = 0.
Proof. We know that qn(x) = det[circ(c1, . . . , cn)], and since we are over C
here we know that qn(x) is the product of the eigenvalues of circ(c1, . . . , cn),
which are given, as we mentioned before, by {pC(ω
j−1)}, for all 1 ≤ j ≤ n,
where pC(X) = c1 + c2X + . . .+ cnX
n−1 and ω = cos(2π
n
) + isin(2π
n
), so
the result follows.
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H. Albuquerque, F. Panaite 5
We will present explicit formulae for the forms qi for n = 3 and n = 4.
The case n = 3 was studied in detail in [1], where the formulae for q1, q2, q3
have been found. Namely, for x = c1e1 + c2e2 + c3e3 ∈ KZ3, we have:
q1(x) = 3c1, q2(x) = 3c21 − 3c2c3, q3(x) = c31 + c32 + c33 − 3c1c2c3.
Moreover, the following properties have been found in [1]:
Proposition 2.7. If x, y ∈ KZ3, then q1(x) = q2(x) and q2(x + y) =
q2(x) + q2(y) + q1(x)q1(y)− q1(xy).
Similarly, by performing explicit computations, one can show that for
x = c1e1 + c2e2 + c3e3 + c4e4 ∈ KZ4, we have:
q1(x) = 4c1,
q2(x) = 6c21 − 4c2c4 − 2c23,
q3(x) = 4c31 − 8c1c2c4 − 4c1c
2
3 + 4c22c3 + 4c3c
2
4,
q4(x) = c41 − c42 + c43 − c44 − 2c21c
2
3 − 4c21c2c4+
+ 4c1c
2
2c3 + 4c1c3c
2
4 + 2c22c
2
4 − 4c2c
2
3c4,
and also by explicit computations using these formulae one can prove:
Proposition 2.8. If x, y ∈ KZ4, then q1(x) = q3(x) and q2(x + y) =
q2(x) + q2(y) + q1(x)q1(y)− q1(xy).
Actually, the above results admit generalizations to arbitrary n, at
least over C:
Proposition 2.9. If x ∈ CZn, then q1(x) = qn−1(x).
Proof. We denote by s0, s1, s2, . . . , sn the elementary symmetric polyno-
mials in n variables X1, X2, . . . , Xn, defined by s0 = 1, s1 = X1+ . . .+Xn,
s2 =
∑
i<j
XiXj , . . . , sn = X1X2 . . . Xn. Also, for any 1 ≤ k ≤ n, we con-
sider the polynomial pk defined by pk = Xk
1 + Xk
2 + . . . + Xk
n. These
polynomials are related by Newton’s identities:
ksk =
k
∑
i=1
(−1)i−1sk−ipi, ∀ 1 ≤ k ≤ n.
In particular, for k = n− 1, we have
(n− 1)sn−1 =
n−1
∑
i=1
(−1)i−1sn−1−ipi. (2.2)
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6 Some algebraic properties of circulant matrices
Let now x = c1e1 + . . .+ cnen ∈ CZn and consider the circulant matrix
circ(c1, . . . , cn), whose eigenvalues will be denoted by λ1, . . . , λn. It is clear
that, for any 1 ≤ i ≤ n, we have qi(x) = si(λ1, . . . , λn). On the other hand,
for any 1 ≤ k ≤ n, the eigenvalues of the circulant matrix associated to
xk are λk1, . . . , λ
k
n and so we have q1(x
k) = λk1 + . . .+ λkn = pk(λ1, . . . , λn).
Now we compute, directly from the formula (2.1):
q1(x) = (−1)n+1q1(x
n−1) + (−1)nq1(x)q1(x
n−2)+
. . .− qn−2(x)q1(x) + nqn−1(x)
= (−1)n+1pn−1(λ1, . . . , λn) + (−1)ns1(λ1, . . . , λn)pn−2(λ1, . . . , λn)+
. . .+ sn−3(λ1, . . . , λn)p2(λ1, . . . , λn)− sn−2(λ1, . . . , λn)p1(λ1, . . . , λn)+
+ nsn−1(λ1, . . . , λn)
(2.2)
= sn−1(λ1, . . . , λn) = qn−1(x),
finishing the proof.
Proposition 2.10. If x, y ∈ CZn, then q2(x + y) = q2(x) + q2(y) +
q1(x)q1(y)− q1(xy).
Proof. We denote by λ1, . . . , λn (respectively µ1, . . . , µn) the eigenvalues
of the circulant matrix corresponding to x (respectively y). Then the
eigenvalues of the circulant matrix corresponding to x+ y (respectively
xy) are λ1 + µ1, . . . , λn + µn (respectively λ1µ1, . . . , λnµn). So, we have:
q2(x+ y) =
∑
i<j
(λi + µi)(λj + µj)
=
∑
i<j
λiλj +
∑
i<j
µiµj +
∑
i<j
λiµj +
∑
i<j
µiλj
= q2(x) + q2(y) +
∑
i<j
λiµj +
∑
i<j
µiλj ,
q1(x)q1(y)− q1(xy) = (
n
∑
i=1
λi)(
n
∑
j=1
µj)−
n
∑
k=1
λkµk
=
∑
i<j
λiµj +
∑
i<j
µiλj ,
hence indeed we have q2(x+ y) = q2(x) + q2(y) + q1(x)q1(y)− q1(xy).
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H. Albuquerque, F. Panaite 7
3. Hopf-algebraic properties of circulant matrices
Let H be a finite dimensional Hopf algebra. We denote as usual by
⇀ the left regular action of H on H∗ defined by (h ⇀ ϕ)(h′) = ϕ(h′h),
for all h, h′ ∈ H and ϕ ∈ H∗. It is well known that with this action H∗
becomes a left H-module algebra, so we can consider the smash product
H∗#H (which is sometimes called in the literature the Heisenberg double
of H∗). It is also well known (see [4], p. 162) that H∗#H is isomorphic
as an algebra to the endomorphism algebra End(H∗), an explicit algebra
isomorphism being defined by
λ : H∗#H ≃ End(H∗), λ(ϕ#h)(ψ) = ϕ ∗ (h ⇀ ψ),
for all ϕ, ψ ∈ H∗ and h ∈ H, where ∗ is the convolution product in H∗
defined by (ϕ ∗ ψ)(h) = ϕ(h1)ψ(h2), where we used the Sweedler-type
notation ∆(h) = h1 ⊗ h2 for the comultiplication of H. In particular,
the restrictions of λ to H∗ and H define embeddings of H∗ and H into
End(H∗) (as algebras), defined respectively by
λH∗ : H∗ → End(H∗), λH∗(ϕ)(ψ) = ϕ ∗ ψ,
λH : H → End(H∗), λH(h)(ψ) = h ⇀ ψ.
If we identify End(H∗) with Mn(K), where n = dim(H), we obtain
algebra embeddings of H and H∗ into Mn(K). Our aim is to see how these
embeddings look like if we take H = KZn with its usual Hopf algebra
structure.
Proposition 3.1. The image of (KZn)
∗ in Mn(K) via the above embed-
ding is the algebra Dn
K of diagonal n× n matrices. The image of KZn in
Mn(K) via the above embedding is the algebra Cn
K of circulant matrices and
the embedding λKZn
coincides with the algebra isomorphism g : KZn ≃ Cn
K
defined in Theorem 2.1.
Proof. We consider the basis {e1, . . . , en} in KZn as before and its dual
basis {p1, . . . , pn} in (KZn)
∗, defined by pi(ej) = δij for all i, j ∈ {1, . . . , n}.
Thus, the algebra structure of (KZn)
∗ is defined on this basis by pipj =
δijpj and p1 + . . .+ pn = 1. Moreover, we identify End((KZn)
∗) ≡Mn(K)
via this basis, that is, if f ∈ End((KZn)
∗) and f(pj) =
∑n
i=1 aijpi, we
identify f with the matrix Mf = (aij)1≤i,j≤n.
By using the formula for λH∗ given above, we can easily see that
λ(KZn)∗(pk)(pj) = pkpj = δjkpk, for all 1 ≤ j, k ≤ n, that is λ(KZn)∗(pk)
coincides via the identification End((KZn)
∗) ≡ Mn(K) with the matrix
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8 Some algebraic properties of circulant matrices
having 1 in the (k, k) position and 0 elsewhere. Thus, for an arbitrary
element x = d1p1 + . . . + dnpn ∈ (KZn)
∗, with d1, . . . , dn ∈ K, we have
Mλ(KZn)∗ (x) = diag(d1, . . . , dn), the diagonal matrix with entries d1, . . . , dn,
q.e.d.
On the other hand, by using the formula for λH given above, we
can see that λKZn
(ek)(pj) = ek ⇀ pj , for all 1 ≤ j, k ≤ n, and since
we have (ek ⇀ pj)(ei) = pj(eiek) = pj(ek+i−1) = δj,k+i−1, for all 1 ≤
i, j, k ≤ n, we obtain ek ⇀ pj = pj−k+1(mod n), that is λKZn
(ek)(pj) =
pj−k+1(mod n), for all 1 ≤ j, k ≤ n, and this means exactly thatMλKZn (ek)
=
circ(0, 0, . . . , 1, . . . , 0), where 1 is in the kth position. Thus, for an arbitrary
element y = c1e1 + . . . + cnen ∈ KZn, with c1, . . . , cn ∈ K, we have
MλKZn (y)
= circ(c1, . . . , cn), q.e.d.
Let us recall some facts from [2]. If X is an (associative unital) algebra
(with multiplication denoted by x ⊗ y 7→ xy for all x, y ∈ X) and A, B
are subalgebras of X, we say that X factorizes as X = AB if the map
A ⊗ B → X, a ⊗ b 7→ ab, for all a ∈ A, b ∈ B, is a linear isomorphism.
This is equivalent to saying that there exists a so called twisting map
R : B ⊗ A → A ⊗ B such that X is isomorphic as an algebra to the so
called twisted tensor product A⊗R B.
It is well known that any smash product (such as (KZn)
∗#KZn) is
a particular case of a twisted tensor product. Since (KZn)
∗ ≃ Dn
K and
KZn ≃ Cn
K as algebras, we can conclude:
Proposition 3.2. The algebra of n× n matrices factorizes as Mn(K) =
Dn
KC
n
K.
Remark 3.3. Assume that K = C. In this case, it is known that the
Hopf algebra CZn is selfdual, that is CZn is isomorphic as a Hopf algebra
to (CZn)
∗. If we consider ω = cos(2π
n
) + isin(2π
n
), an explicit isomor-
phism φ : CZn ≃ (CZn)
∗ is defined by φ(ei) =
∑n
j=1 ω
(i−1)(j−1)pj , for
all i ∈ {1, . . . , n}, where {e1, . . . , en} is the standard basis in CZn and
{p1, . . . , pn} is its dual basis in (CZn)
∗. We consider the algebra isomor-
phisms CZn ≃ Cn
C and (CZn)
∗ ≃ Dn
C and thus we obtain an algebra
isomorphism ψ : Cn
C ≃ Dn
C. If C = circ(c1, . . . , cn) is a circulant ma-
trix, an easy computation shows that ψ(C) = diag(λ1, . . . , λn), where
{λ1, . . . , λn} are the eigenvalues of C defined by λj = c1 + c2ω
j−1 +
c3ω
2(j−1) + . . .+ cnω
(n−1)(j−1), for all j ∈ {1, . . . , n}. In particular, this
shows immediately the known fact that if X,Y are circulant matrices
with eigenvalues λ1, . . . , λn and respectively β1, . . . , βn, then the circulant
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H. Albuquerque, F. Panaite 9
matrix X + Y has eigenvalues λ1 + β1, . . . , λn + βn and the circulant
matrix XY has eigenvalues λ1β1, . . . , λnβn.
We denote by Pn the fundamental circulant n × n matrix, defined
by Pn = circ(0, 1, 0, . . . , 0). It has the property that Pn
n = In and, if
C = circ(c1, . . . , cn), then C = c1In + c2Pn + c3P
2
n + . . .+ cnP
n−1
n . If we
denote as before by {e1, . . . , en} the standard basis of KZn, we obviously
have e22 = e3, e
3
2 = e4, . . . , en−1
2 = en. Thus, since g is an algebra map
and g(e2) = Pn, we have, for all 2 ≤ i ≤ n, g(ei) = g(ei−1
2 ) = P i−1
n . Recall
also that, for any 1 ≤ i ≤ n, the inverse of ei in KZn is en−i+2.
We look again at the algebra isomorphism g : KZn ≃ Cn
K. Since KZn
is a Hopf algebra, we can transfer its structure to Cn
K via g, and thus Cn
K
becomes a Hopf algebra. We will write down its counit, comultiplication
and antipode.
Let C = circ(c1, . . . , cn) be a circulant matrix. It is easy to see that
the transferred counit is defined by ε : Cn
K → K, ε(C) = c1 + c2 + . . .+ cn.
We compute now the comultiplication ∆ : Cn
K → Cn
K ⊗ Cn
K:
∆(C) = (g ⊗ g) ◦∆KZn
◦ g−1(C)
= (g ⊗ g) ◦∆KZn
(c1e1 + . . .+ cnen)
= (g ⊗ g)(c1e1 ⊗ e1 + . . .+ cnen ⊗ en)
= c1g(e1)⊗ g(e1) + . . .+ cng(en)⊗ g(en)
= c1In ⊗ In + c2Pn ⊗ Pn + c3P
2
n ⊗ P 2
n + . . .+ cnP
n−1
n ⊗ Pn−1
n .
Note that the counit property (ε⊗ id) ◦∆ = id applied to C becomes, by
using the above formulae and the fact that ε(Pn) = 1,
C = (ε⊗ id) ◦∆(C)
= (ε⊗ id)(c1In ⊗ In + c2Pn ⊗ Pn + c3P
2
n ⊗ P 2
n + . . .+ cnP
n−1
n ⊗ Pn−1
n )
= c1In + c2Pn + c3P
2
n + . . .+ cnP
n−1
n ,
that is, the counit property is equivalent to the basic property of the fun-
damental circulant matrix. We compute now the formula for the antipode
S : Cn
K → Cn
K:
S(C) = g ◦ SKZn
◦ g−1(C)
= g ◦ SKZn
(c1e1 + . . .+ cnen)
= g(c1e1 + c2e
−1
2 + . . .+ cne
−1
n )
= g(c1e1 + c2en + c3en−1 + . . .+ cne2)
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10 Some algebraic properties of circulant matrices
= circ(c1, cn, cn−1, . . . , c3, c2)
= circ(c1, c2, . . . , cn)
T .
That is, S(C) is just the transpose of C.
If we denote by Qn = circ(0, 0, . . . , 0, 1) = P T
n the transpose of Pn,
then the antipode property S(C(1))C(2) = ε(C)In, with notation ∆(C) =
C(1) ⊗ C(2), is equivalent to the following relation involving the matrices
Pn and Qn:
c1In + c2QnPn + c3Q
2
nP
2
n + . . .+ cnQ
n−1
n Pn−1
n = (c1 + . . .+ cn)In,
which is obviously true because we actually have QnPn = In, that is Qn
is the inverse of Pn.
Remark 3.4. It is well known that the element x = 1
n
(e1 + . . .+ en) is a
so called integral in KZn, that is it satisfies the condition hx = ε(h)x for
all h ∈ KZn. If we write h = c1e1 + . . .+ cnen, with c1, . . . , cn ∈ K, then
the equality hx = ε(h)x may be transferred in Cn
K via the isomorphism g,
and we obtain g(h)g(x) = ε(h)g(x), that is
circ(c1, . . . , cn)circ(1, 1, . . . , 1) = (c1 + . . .+ cn)circ(1, 1, . . . , 1),
and this is equivalent to the fact that c1 + . . .+ cn is an eigenvalue for
circ(c1, . . . , cn) with eigenvector (1, 1, . . . , 1)T .
4. Brandt algebras and lattices in circulat matrices
In this section we assume that the base field is C.
Theorem 4.1. Let C = circ(c1, . . . , cn) be a circulant matrix. We de-
note by ∆ the comultiplication of the Hopf algebra Cn
C and by Pn the
fundamental circulant matrix circ(0, 1, 0, . . . , 0). Then ∆(C), regarded
as an n2 × n2 matrix, is the block circulant with circulant blocks matrix
circ(c1In, c2Pn, . . . , cnP
n−1
n ). Moreover, the eigenvalues of ∆(C) are the
eigenvalues of C, each one with multiplicity n.
Proof. The first statement follows easily from the formula ∆(C) = c1In ⊗
In + c2Pn ⊗ Pn + c3P
2
n ⊗ P 2
n + . . .+ cnP
n−1
n ⊗ Pn−1
n . To prove the second
statement, one verifies first that the block diagonal matrix that diagonalizes
∆(C) is
diag(Λ1 + Λ2 + . . .+ Λn,Λ1 + ωΛ2 + . . .+ ωn−1Λn, . . . ,
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H. Albuquerque, F. Panaite 11
. . . ,Λ1 + ωn−1Λ2 + . . .+ ω(n−1)2Λn),
where ω = cos(2π
n
) + isin(2π
n
), and
Λ1 = c1In, Λ2 = c2diag(1, ω, . . . , ω
n−1), . . . ,
. . .Λn = cndiag(1, ω
n−1, . . . , ω(n−1)2),
and the eigenvalues of ∆(C) are still pC(1), pC(ω), . . . , pC(ω
n−1), each one
with multiplicity n.
Recalling that the algebra of complex circulant matrices is isomorphic
to CZn, we can introduce the following analogue of the concept in [1]:
Definition 4.2. A set B of complex circulant matrices is called an integral
(rational) Brandt algebra if qi(a), qi(b), qi(a + b), qi(ab) ∈ Z(Q), for all
a, b ∈ B and 1 ≤ i ≤ n, where qi are the forms defined in Corollary 2.2,
transferred to Cn
C as in Remark 2.3.
Proposition 4.3. The set of complex circulant matrices that have integral
(rational) eigenvalues is an integral (rational) Brandt algebra.
Proof. Follows immediately from the fact that the eigenvalues of a sum
(product) of circulant matrices are the sums (products) of the eigenvalues
of the given matrices and by using the fact that for a circulant matrix C
all qi(C) are symmetric polynomials in the eigenvalues of C.
Proposition 4.4. The set of complex circulant matrices C = circ(c1, . . . , cn)
that have integer (rational) eigenvalues is given by
(c1, c2, . . . , cn)
T =
1
n
M(λ1, . . . , λn)
T ,
where λ1, . . . , λn are integer (rational) numbers, and the n × n matrix
M has ω(i−1)(j−1) as the (i, j) entry, where ω = cos(2π
n
) + isin(2π
n
).
Moreover, if the elements λi satisfy the extra condition λk+1 = λn−k+1,
for all 1 ≤ k ≤ n− 1, then the matrix C has real entries.
Proof. If we consider the matrix A = (aij) such that aij = ω(i−1)(j−1), it
is easy to see that A(c1, . . . , cn)
T = (pC(1), pC(ω), . . . , pC(ω
n−1)), and the
result follows because the inverse of A is 1
n
M . For the second statement,
just note that in the matrix M , the column k + 1 is the conjugate of the
column n− k + 1.
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12 Some algebraic properties of circulant matrices
Corollary 4.5. The set of block circulants with circulant blocks ∆(B),
where B is the set of circulant matrices with integral (rational) eigenval-
ues and ∆ is the comultiplication of the Hopf algebra Cn
C, is an integral
(rational) Brandt algebra.
Proof. The result follows because, as we have seen in Theorem 4.1, the
eigenvalues of ∆(b), for any b ∈ B, coincide with the eigenvalues of b.
Now we can study lattices in the algebra of complex circulant matrices.
This study takes us to some conditions about when a circulant matrix is
obtained as integral linear combinations of some elements of this algebra.
Here we will focus to the subset of circulant matrices that have integral
entries.
Theorem 4.6. We consider the lattice Zv1+Zv2+ . . .+Zvn in the algebra
of complex circulant n× n matrices, generated by the linearly independent
vectors
v1 = c11In + c12Pn + . . .+ c1nP
n−1
n ,
v2 = c21In + c22Pn + . . .+ c2nP
n−1
n ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vn = cn1In + cn2Pn + . . .+ cnnP
n−1
n .
We denote by C the n× n matrix with entries cij. If all the entries of the
matrix C−1 are integral, then any circulant matrix with integral entries
belongs to this lattice, that is, may be written as a1v1 + a2v2 + . . .+ anvn,
with a1, . . . , an ∈ Z.
Proof. The matrix C is invertible because we assumed that v1, . . . , vn
are linearly independent. Since we assumed that the entries of C−1 are
integral, it follows that each of the matrices In, Pn, P 2
n , . . . , Pn−1
n may be
written as an integral linear combination of the elements v1, . . . , vn, and
now the result follows because any circulant matrix circ(c1, . . . , cn) may
be written as circ(c1, . . . , cn) = c1In + c2Pn + c3P
2
n + . . .+ cnP
n−1
n .
Moreover, if we consider the lattice Z∆(v1) + Z∆(v2) + . . .+ Z∆(vn)
in the set of block circulants with circulant blocks, if C−1 has inte-
gral entries then every block circulant with circulant blocks of the form
circ(a1In, a2Pn, . . . , anP
n−1
n ), with a1, . . . , an ∈ Z belongs to this lattice,
that is may be written as an integral linear combination of ∆(v1), . . . ,∆(vn).
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H. Albuquerque, F. Panaite 13
Example 4.7. Consider the lattice of 3×3 circulant matrices Zv1+Zv2+
Zv3, with
v1 = −P3 + P 2
3 ,
v2 = −
1
3
I3 +
1
3
P3 +
1
3
P 2
3 ,
v3 =
1
3
I3 +
2
3
P3 −
1
3
P 2
3 .
The matrix of coefficients
0 −1 1
−1/3 1/3 1/3
1/3 2/3 −1/3
has an inverse with integral entries, so every circulant 3× 3 matrix with
integral entries may be written as av1+bv2+cv3, with a, b, c ∈ Z. Moreover,
each matrix circ(a1I3, a2P3, a3P
2
3 ), with a1, a2, a3 ∈ Z, may be written as
m1∆(v1) +m2∆(v2) +m3∆(v3), with m1,m2,m3 ∈ Z.
5. A class of generalized circulants
There exist various generalizations of circulant matrices, appeared
around 1980, see for instance [3], [5] and references therein. As proved by
Waterhouse in [5], many of them are related to twisted group rings. In
this section we will introduce a certain class of matrices that generalize
both circulant and skew circulant matrices, also related to twisted group
rings, having the property that their eigenvalues can be read directly from
the entries of the matrices.
Consider again the cyclic group Zn with elements denoted as before
by e1, . . . , en, and F : Zn × Zn → K∗ a two-cocycle (here K∗ is the set of
nonzero elements in K), that is F satisfies
F (e1, x) = F (x, e1) = 1, ∀ x ∈ Zn,
F (x, y)F (xy, z) = F (y, z)F (x, yz), ∀ x, y, z ∈ Zn.
We can consider then the twisted group ring KFZn, which is an associative
algebra (with unit e1) obtained from the group ring KZn by deforming
its product using F , namely x ·F y = F (x, y)xy, for all x, y ∈ Zn.
We have seen that KZn may be embedded as an algebra in Mn(K)
(and the image of this embedding is the algebra of circulant matrices Cn
K).
We can do something similar for a twisted group ring KFZn. Namely,
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14 Some algebraic properties of circulant matrices
define the map λF : KFZn → End((KFZn)
∗), λF (a)(ψ) = a ⇀ ψ, where
(a ⇀ ψ)(b) = ψ(ba), for all a, b ∈ KFZn and ψ ∈ (KFZn)
∗. If we denote by
{pj} the basis of (KFZn)
∗ dual to the basis {ej} of KFZn, one can check
that we have λF (ek)(pj) = F (ej−k+1, ek)pj−k+1, for all j, k ∈ {1, . . . , n}.
If we identify as usual End((KFZn)
∗) ∼= Mn(K), we obtain an algebra
embedding λF : KFZn →֒ Mn(K), defined as follows: if x = c1e1 +
. . .+ cnen ∈ KFZn, then λF (x) is the n× n matrix whose (i, j) entry is
cj−i+1F (ei, ej−i+1). For instance, for n = 3 and n = 4 the corresponding
matrices are respectively given by
c1 c2 c3
c3F (e2, e3) c1 c2F (e2, e2)
c2F (e3, e2) c3F (e3, e3) c1
c1 c2 c3 c4
c4F (e2, e4) c1 c2F (e2, e2) c3F (e2, e3)
c3F (e3, e3) c4F (e3, e4) c1 c2F (e3, e2)
c2F (e4, e2) c3F (e4, e3) c4F (e4, e4) c1
We denote by Cn
K(F ) the image of the map λF ; so Cn
K(F ) is an algebra,
isomorphic to KFZn.
Assume now that the two-cocycle F is trivial in the cohomology group
H2(Zn,K∗), that is there exists a map µ : Zn → K∗, with µ(e1) = 1 (and
with notation µ(ei) = µi for all i ∈ {2, . . . , n}), such that F (ei, ej) =
µ(ei)µ(ej)µ(eiej)
−1, for all i, j ∈ {1, . . . , n}. We will denote the algebra
Cn
K(F ) by Cn
K(µ). Also, for x = c1e1 + . . .+ cnen ∈ KFZn, we will denote
λF (x) := λµ(x) := circ(c1, c2, . . . , cn;µ2, . . . , µn),
which is a matrix having c1, c2, . . . , cn in the first row, c1 on the main diag-
onal and entry cj−i+1
µiµj−i+1
µj
in any other position (i, j) (with the conven-
tion µ1 = 1). Obviously, we have circ(c1, . . . , cn) = circ(c1, . . . , cn; 1, . . . , 1).
For instance, for n = 3, the matrix circ(c1, c2, c3; a, b) is
c1 c2 c3
c3ab c1 c2a
2b−1
c2ab c3b
2a−1 c1
We aim to find the eigenvalues and eigenvectors for a matrix
circ(c1, c2, . . . , cn;µ2, . . . , µn). For this, we will rely on the known fact
that for the two-cocycle F given by F (ei, ej) = µ(ei)µ(ej)µ(eiej)
−1 the
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twisted group ring KFZn is isomorphic to the group ring KZn, an iso-
morphism being defined by ϕ : KFZn ≃ KZn, ϕ(ei) = µ(ei)ei for all
1 ≤ i ≤ n. Thus, we have also an algebra isomorphism Ψ : Cn
K(µ) ≃ Cn
K
defined by Ψ(circ(c1, . . . , cn;µ2, . . . , µn)) = circ(c1, c2µ2, c3µ3, . . . , cnµn)
with inverse Ψ−1(circ(c1, . . . , cn)) = circ(c1,
c2
µ2
, . . . , cn
µn
;µ2, . . . , µn).
Proposition 5.1. Assume that λ is an eigenvalue for the ma-
trix circ(c1, c2µ2, . . . , cnµn), with eigenvector (x1, x2, . . . , xn)
T . Then
λ is an eigenvalue for circ(c1, c2, . . . , cn;µ2, . . . , µn) with eigenvector
(x1, x2µ2, . . . , xnµn)
T .
Proof. In order to be able to use the algebra isomorphism Ψ, we need to
transform the equality
circ(c1, c2µ2, . . . , cnµn)(x1, . . . , xn)
T = λ(x1, . . . , xn)
T (5.1)
into a relation between circulant matrices. We consider the
circulant circ(x1, xn, xn−1, . . . , x3, x2), and we remark that the
first column of circ(c1, c2µ2, . . . , cnµn)circ(x1, xn, xn−1, . . . , x3, x2)
is exactly circ(c1, c2µ2, . . . , cnµn)(x1, . . . , xn)
T , and by (5.1)
this column is (λx1, . . . , λxn)
T . On the other hand,
circ(c1, c2µ2, . . . , cnµn)circ(x1, xn, xn−1, . . . , x3, x2) is a circu-
lant matrix (being the product of two circulants) and since
we know that its first column is (λx1, . . . , λxn)
T we find
out that circ(c1, c2µ2, . . . , cnµn)circ(x1, xn, xn−1, . . . , x3, x2) =
circ(λx1, λxn, λxn−1, . . . , λx3, λx2). To this equality we apply the
algebra map Ψ−1; we obtain:
circ(c1, c2, . . . , cn;µ2, . . . , µn)circ(x1,
xn
µ2
,
xn−1
µ3
, . . . ,
x3
µn−1
,
x2
µn
;µ2, . . . , µn)
= circ(λx1,
λxn
µ2
,
λxn−1
µ3
, . . . ,
λx3
µn−1
,
λx2
µn
;µ2, . . . , µn).
The first column of the matrix circ(x1,
xn
µ2
, xn−1
µ3
, . . . , x3
µn−1
, x2
µn
;µ2, . . . , µn)
is
(x1,
x2
µn
F (e2, en),
x3
µn−1
F (e3, en−1), . . . ,
xn−1
µ3
F (en−1, e3),
xn
µ2
F (en, e2))
T ,
that is
(x1,
x2
µn
µ2µn,
x3
µn−1
µ3µn−1, . . . ,
xn−1
µ3
µn−1µ3,
xn
µ2
µnµ2)
T ,
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16 Some algebraic properties of circulant matrices
which is (x1, x2µ2, . . . , xn−1µn−1, xnµn)
T . Similarly, we can see that the
first column of the matrix circ(λx1,
λxn
µ2
, λxn−1
µ3
, . . . , λx3
µn−1
, λx2
µn
;µ2, . . . , µn)
is (λx1, λx2µ2, . . . , λxnµn)
T . So, the above matrix equality im-
plies circ(c1, . . . , cn;µ2, . . . , µn)(x1, x2µ2, . . . , xn−1µn−1, xnµn)
T =
λ(x1, x2µ2, . . . , xnµn)
T , i.e. λ is an eigenvalue for
circ(c1, c2, . . . , cn;µ2, . . . , µn) with eigenvector (x1, x2µ2, . . . , xnµn)
T .
Assume now that K = C. Since we know ([3]) the eigenvalues and eigen-
vectors for any circulant matrix, in particular for circ(c1, c2µ2, . . . , cnµn),
we obtain immediately from this Proposition the eigenvalues and eigen-
vectors for circ(c1, c2, . . . , cn;µ2, . . . , µn):
Proposition 5.2. For the complex matrix C =
circ(c1, c2, . . . , cn;µ2, . . . , µn) define the polynomial pC(X) =
c1 + c2µ2X + c3µ3X
2 + . . . + cnµnX
n−1. For j ∈ {1, 2, . . . , n}
define λj = pC(ω
j−1), where ω = cos(2π
n
) + isin(2π
n
). Then
λ1, . . . , λn are the eigenvalues of C, and the eigenvector of λj is
xj = (1, µ2ω
j−1, µ3ω
2(j−1), . . . , µnω
(n−1)(j−1))T , for all 1 ≤ j ≤ n.
Remark 5.3. Exactly as for ordinary circulant matrices, it follows
that every element x = c1e1 + . . . + cnen ∈ KFZn is a solu-
tion to a certain polynomial Xn − q1(x)X
n−1 + . . . + (−1)nqn(x),
with q1(x) = tr[circ(c1, . . . , cn;µ2, . . . , µn)] = nc1 and qn(x) =
det[circ(c1, . . . , cn;µ2, . . . , µn)]. Moreover, if K = C, then for any 1 ≤
i ≤ n we have that qi(x) = si(λ1, . . . , λn), where si is the ith ele-
mentary symmetric polynomial and λ1, . . . , λn are the eigenvalues of
circ(c1, . . . , cn;µ2, . . . , µn).
We show now that, over the field C, the matrices of the type
circ(c1, c2, . . . , cn;µ2, . . . , µn) generalize not only circulant matrices but
also skew circulant matrices. Recall from [3] that a skew circulant matrix
is a circulant followed by a change in sign to all the elements below the
main diagonal. Such a matrix is denoted by scirc(c1, . . . , cn). For example,
scirc(a, b, c) is the matrix
a b c
−c a b
−b −c a
For a given n, we denote by σ = cos(π
n
) + isin(π
n
) and ω = σ2 =
cos(2π
n
) + isin(2π
n
). With this notation, a straightforward computation
(using the fact that σn = −1) shows:
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Proposition 5.4. scirc(c1, . . . , cn) = circ(c1, . . . , cn;σ, σ
2, . . . , σn−1).
Consequently, the skew circulant matrices are a subalgebra of Mn(C)
(this was noticed also in [3]), which will be denoted by sCn
C. By what we
have done before it follows that we have an algebra isomorphism Ψ : sCn
C ≃
Cn
C, Ψ(scirc(c1, . . . , cn)) = circ(c1, σc2, σ
2c3, . . . , σ
n−1cn). Moreover, the
eigenvalues of a skew circulant matrix, computed in [3], may be reobtained
by applying Proposition 5.2: namely, the eigenvalues of scirc(c1, . . . , cn) =
circ(c1, . . . , cn;σ, σ
2, . . . , σn−1) are given by λj = pC(ω
j−1), for 1 ≤ j ≤ n,
where pC(X) = c1 + c2σX + c3σ
2X2 + . . .+ cnσ
n−1Xn−1.
References
[1] H. Albuquerque, R. S. Krausshar, Multiplicative invariant lattices in Rn obtained
by twisting of group algebras and some explicit characterizations, J. Algebra 319
(2008), 1116–1131.
[2] A. Cap, H. Schichl, J. Vanžura, On twisted tensor products of algebras, Comm.
Algebra 23 (1995), 4701–4735.
[3] P. J. Davis, Circulant Matrices, John Wiley, New York, 1977.
[4] S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Reg. Conf.
Series 82, Providence, RI, 1993.
[5] J. C. Waterhouse, The structure of monomial circulant matrices, SIAM J. Alg.
Disc. Meth. 8 (1987), 467–482
Contact information
H. Albuquerque Departamento de Matemática, Universidade de
Coimbra, 3001-454 Coimbra, Portugal
E-Mail: lena@mat.uc.pt
F. Panaite Institute of Mathematics of the Romanian
Academy, PO-Box 1-764, RO-014700 Bucharest,
Romania
E-Mail: Florin.Panaite@imar.ro
Received by the editors: 03.12.2011
and in final form 03.12.2011.
H. Albuquerque, F. Panaite
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