- Email: [email protected]
The product of an involution and a skew-involution
Linear Algebra and its Applications 584 (2020) 431–437
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
The product of an involution and a skew-involution Agnes T. Paras ∗ , Jenny R. Salinasan Institute of Mathematics, University of the Philippines, Diliman, Quezon City 1101, Philippines
a r t i c l e
i n f o
Article history: Received 18 September 2019 Accepted 25 September 2019 Available online 30 September 2019 Submitted by V.V. Sergeichuk
a b s t r a c t We give necessary and sufficient conditions for a matrix over a field to be the product of an involution (A2 = I) and a skew-involution (A2 = −I). © 2019 Elsevier Inc. All rights reserved.
MSC: 15A21 15A23 Keywords: Involution Skew-involution Companion matrix Frobenius canonical form
1. Introduction Let Mn (F) denote the set of all n-by-n matrices over a field F . Suppose A ∈ Mn (F ). The matrix A is said to be an involution if A2 = I. Wonenburger proved in 1966 that if the characteristic of F is not equal to 2, then A is a product of two involutions if and only if A is similar to A−1 [5]. A year after, Djoković gave a proof of the preceding result for an arbitrary field [1]. Ellers showed in 1979 that the result fails to hold when F is a * Corresponding author. E-mail addresses: [email protected] (A.T. Paras), [email protected] (J.R. Salinasan). https://doi.org/10.1016/j.laa.2019.09.035 0024-3795/© 2019 Elsevier Inc. All rights reserved.
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A.T. Paras, J.R. Salinasan / Linear Algebra and its Applications 584 (2020) 431–437
noncommutative field [2]. In 2008, Furtado gave necessary and sufficient conditions for A to be a product of two involutions with prescribed eigenvalues [3]. We say that B ∈ Mn (F) is a skew-involution if B 2 = −I. If the characteristic of F is two, then B is a skew-involution if and only if B is an involution. We wish to find necessary and sufficient conditions for A ∈ Mn (F) to be a product of an involution and a skew-involution. Suppose the polynomial x2 + 1 has a root u in F . Let A ∈ Mn (F ) be given. Notice that A is a skew-involution if and only if uA is an involution. Hence, A = ST , where S is an involution and T is a skew-involution if and only if uA = S(uT ) is a product of two involutions. By the theorem of Wonenburger and Djoković, the latter statement is equivalent to uA is similar to (uA)−1 = −uA−1 . Therefore if x2 + 1 has a root in F , then A is a product of an involution and a skew-involution if and only if A is similar to −A−1 . We show that this is also the case when x2 + 1 has no root in F . 2. The Frobenius canonical form of −A−1 Denote by GLn (F) the set of all nonsingular matrices in Mn (F ). Let F [x] be the set of all polynomials in x over F , and denote the minimal polynomial of A ∈ Mn (F ) by mA (x). n Let h(x) be a monic polynomial over F with positive degree n, say h(x) = ai xi . The i=0
Frobenius block or companion matrix of h(x) is defined to be ⎡
0 ··· ⎢ ⎢1 C(h) = ⎢ .. ⎢ . ⎣
⎤ −a0 ⎥ −a1 ⎥ .. ⎥ ⎥ ∈ Mn (F ). . ⎦ 1 −an−1 0
It is known that the minimal and characteristic polynomials of C(h) are both equal to h(x). The Frobenius (or rational) canonical form theorem states that if A ∈ Mn (F ), then A is similar to a direct sum of Frobenius blocks ⊕ki=1 C(pei i ), where each pi (x) ∈ F [x] irreducible and each ei is a positive integer. The direct sum ⊕ki=1 C(pei i ) is unique up to permutation of summands. The polynomials pi (x)ei for i = 1, . . . , k are called the elementary divisors of A ([4], Corollary 4.7). Thus two matrices in Mn (F ) are similar if and only if they have the same elementary divisors. In particular, if x2 + 1 has no root in
0 −I n2 F and T ∈ Mn (F) is a skew-involution, then n is even and T is similar to , I n2 0 since mT (x) = x2 + 1 is irreducible in F [x]. An immediate consequence of the Frobenius canonical form theorem is the following. Corollary 1. Let A ∈ Mn (F) and B, C ∈ Mm (F) be given. If A ⊕ B is similar to A ⊕ C, then B is similar to C.
A.T. Paras, J.R. Salinasan / Linear Algebra and its Applications 584 (2020) 431–437
433
Let A, B ∈ GLn (F) such that A = T −1 BT for some T ∈ GLn (F ). Suppose there is an involution P ∈ GLn (F) such that P −1 AP = −A−1 . If Q := T P T −1 , then Q2 = I and Q−1 BQ = T P −1 T −1 BT P T −1 = T P −1 AP T −1 = T (−A−1 )T −1 = −B −1 . Lemma 2. Suppose A, B ∈ GLn (F) such that A is similar to B. If there exists an involution P ∈ GLn (F) such that P −1 AP = −A−1 , then there exists an involution Q ∈ GLn (F ) such that Q−1 BQ = −B −1 . Suppose R is the Frobenius canonical form of A ∈ GLn (F ). Lemma 2 implies that if we wish to determine whether A is similar to −A−1 via an involution, then it suffices to determine whether R is similar to −R−1 via an involution. Let FM [x] := {g(x) ∈ F [x] | g(x) is monic of positive degree and g(0) = 0}. If g(x) =
n
ai xi ∈ F [x] is of degree n, then g(x) ∈ FM [x] if and only if n ∈ Z+ , an = 1,
i=0
and a0 = 0. Define the degree-preserving function γ : FM [x] → FM [x] by γ(
n
n
r i xi ) =
i=0
i=0
(−1)n−i r0−1 rn−i xi .
Then γ(g(x)) = g(x) if and only if r0 = (−1)n r0−1 and ri = (−1)n−i r0−1 rn−i for 0 < i < n, which is equivalent to r02 = (−1)n and ri = (−1)i r0 rn−i for 0 < i < n. Thus if γ(g(x)) = g(x) and x2 + 1 has no root in F , then n is even and r0 = ±1. We consider other properties of γ in the following lemma. Lemma 3. Let g(x), q(x) ∈ FM [x] of degree n be given. (a) If g(x) =
n i=0
ri xi , then γ(g(x)) = g(x) if and only if r02 = (−1)n and ri =
(−1)i r0 rn−i for 0 < i < n. (b) If γ(q(x)) = g(x), then γ(g(x)) = q(x), that is γ 2 = idFM [x] . (c) The function γ is multiplicative. (d) If g(x) is irreducible, then γ(g(x)) is irreducible. Proof. Let g(x) =
n
ri xi , q(x) =
i=0
n
si xi ∈ FM [x] such that γ(q(x)) = g(x). Then
i=0
n −1 (−1)n−i s−1 and sn−i = 0 sn−i = ri for 0 ≤ i < n, which is equivalent to s0 = (−1) r0 i −1 n −1 n−i −1 (−1) r0 ri for 0 < i < n. This implies s0 = (−1) r0 and si = (−1) r0 rn−i for 0 < i < n, that is q(x) = γ(g(x)). This proves (b).
To show (c), let v(x) =
m i=0
ai xi ∈ FM [x] have degree m. Then
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A.T. Paras, J.R. Salinasan / Linear Algebra and its Applications 584 (2020) 431–437
γ(v(x))γ(g(x)) =(
m j=0
=
j (−1)m−j a−1 0 am−j x )(
m+n
n
k=0
(
(−1)n−k r0−1 rn−k xk )
(−1)m+n−j−k (a0 r0 )−1 am−j rn−k )xi
i=0 j+k=i
=
m+n
(−1)m+n−i (a0 r0 )−1 (
i=0
= γ[
aj rk )xi
j+k=m+n−i
m+n
(
aj rk )xi ] = γ(v(x)g(x)).
i=0 j+k=i
Finally, suppose that γ(g(x)) is reducible, i.e., there exist non-constant f (x), h(x) ∈ F [x] such that γ(g(x)) = f (x)h(x). Note that f (x) and h(x) can be chosen to be monic; if not, take f1 (x) = r−1 f (x) and h1 (x) = s−1 h(x), where r and s are the leading coefficients of f (x) and h(x), respectively, so that γ(g(x)) = f1 (x)h1 (x). Moreover, the constant term of γ(g(x)) is nonzero, so the constant terms of f (x) and h(x) are nonzero. Hence, we can assume without loss of generality that f (x), h(x) ∈ FM [x]. By (b) and (c), g(x) = γ(γ(g(x))) = γ(f (x)h(x)) = γ(f (x))γ(h(x)), that is g(x) is reducible. This proves the contrapositive of (d). 2 k If g(x) = i=1 pi (x)li is a factorization of g(x) ∈ FM [x] into monic, irreducible k polynomials in F [x], then, by Lemma 3, γ(g(x)) = i=1 γ(pi (x))li is a factorization of γ(g(x)) into monic, irreducible polynomials in F [x].
Let Tn = diag −1, 1, −1, 1, . . . , (−1)n−1 , (−1)n ∈ Mn (F ). Denote by Sn = [sij ] the n-by-n backward identity matrix, that is sij = 1, if i + j = n + 1, and sij = 0, otherwise. Observe that (Tn Sn )2 = (−1)n+1 I, that is Tn Sn is an involution when n is odd; Tn Sn is a skew-involution when n is even. It can be readily verified that if g(x) ∈ FM [x] is of degree n, then C(γ(g))Tn Sn C(g) = −Tn Sn , that is −C(g)−1 is similar to C(γ(g)) via Tn Sn . Lemma 4. Let g(x) ∈ FM [x] of degree n be given. (a) (Tn Sn )−1 C(γ(g))Tn Sn = −C(g)−1 . (b) If γ(g(x)) = g(x), then there exists an involution P ∈ GLn (F ) such that P −1 C(g)P = −C(g)−1 . Proof. Suppose g(x) =
n
ri xi ∈ FM [x] is of degree n and γ(g(x)) = g(x). By
i=0
Lemma 3(a), we have r02 = (−1)n and ri = (−1)i r0 rn−i , for 0 < i < n. Define
A.T. Paras, J.R. Salinasan / Linear Algebra and its Applications 584 (2020) 431–437
⎡
1 ⎢ (−1)n r0 r1 ⎢ ⎢ (−1)n r r 0 2 ⎢ ⎢ . ⎢ .. P =⎢ ⎢ n ⎢ (−1) r0 rn−3 ⎢ ⎣ (−1)n r0 rn−2 (−1)n r0 rn−1
0 0 0
0 0 0
0 0 (−1)n r0
0 (−1)n−1 r0 0
··· ··· ···
0 0 0
··· ··· ···
.. 0 0 0
0 0 (−1)3 r0 . 0 0 0
435
⎤ 0 (−1)2 r0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ 0 ⎥ ⎦ 0 0
Then P is an involution, ⎡ ⎢ ⎢ ⎢ P C(g) = ⎢ ⎢ ⎢ ⎣
−r0
0 r0 ..
.
(−1)n−1 r0 n
(−1) r0
⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
0
and [P C(g)]2 = (−1)n+1 r02 I = −I, since r02 = (−1)n and ri = (−1)i r0 rn−i for 0 < i < n. This proves (b). 2 If A ∈ GLn (F) is similar to ⊕ki=1 C(gi ) for some gi (x) ∈ FM [x], then, by Lemma 4(a), −A−1 is similar to ⊕ki=1 C(γ(gi )). 3. Main result We now give necessary and sufficient conditions for A ∈ GLn (F ) to be a product of an involution and a skew-involution. Theorem 5. Let A ∈ GLn (F) be given. The following are equivalent: (i) (ii) (iii) (iv) (v)
A is a product of an involution and a skew-involution. A is similar to −A−1 via an involution. A is similar to −A−1 via a skew-involution. A is similar to −A−1 . The Frobenius canonical form of A is a direct sum of matrices of types (a) C(g) ⊕ C(γ(g)) for some g(x) ∈ FM [x], (b) C(g) for some g(x) ∈ FM [x] such that γ(g(x)) = g(x).
Proof. Let A ∈ GLn (F) be given. (i) ⇐⇒ (ii) ⇐⇒ (iii). If A = ST , where S is an involution and T is a skewinvolution, then S −1 AS = T S = −A−1 , that is A is similar to −A−1 via an involution. If S −1 AS = −A−1 for some involution S, then AS is a skew-involution and (AS)−1 A(AS) = S −1 AS = −A−1 . If K −1 AK = −A−1 for some skew-involution K,
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then AK is an involution and A = (AK)(−K) is a product of an involution and a skew-involution. We have shown that (i), (ii), and (iii) are equivalent. (iii) =⇒ (iv). This is immediate. (iv) =⇒ (v). Suppose that A is similar to −A−1 . Let A be similar to ⊕m i=1 C(gi ), where g1 (x), . . . , gm (x) ∈ FM [x] are the elementary divisors of A. By Lemma 4(a), we m have ⊕m i=1 C(gi ) is similar to ⊕i=1 C(γ(gi )). Without loss of generality, let N ∈ {1, . . . , m} be such that γ(gi (x)) = gi (x) if and only if i > N . By Corollary 1, ⊕N i=1 C(gi ) is similar N to ⊕i=1 C(γ(gi )), where γ(gi (x)) = gi (x) for i ≤ N . If i1 ∈ {1, . . . , N }, then gi1 (x) = γ(gj1 (x)) for some j1 ∈ {1, . . . , N } \ {i1 }. By Lemma 3(b), gj1 (x) = γ(gi1 (x)). Set H1 = {1, . . . , N } \{i1 , j1 }. If H1 = ∅, then N = 2 and A is similar to C(gi1 ) ⊕C(γ(gi1 )) ⊕ ⊕m j=N +1 C(gj ), where γ(gj (x)) = gj (x) for all j > N . If H1 = ∅, then, by Corollary 1, ⊕i∈H1 C(gi ) is similar to ⊕i∈H1 C(γ(gi )). We repeat the above argument to conclude N/2 that N is even and A is similar to ⊕i=1 [C(qi ) ⊕ C(γ(qi ))] ⊕ ⊕m j=N +1 C(gj ) for some qi (x), gj (x) ∈ FM [x] such that γ(qi (x)) = qi (x) for all i and γ(gj (x)) = gj (x) for all j. This proves that (iv) implies (v). (v) =⇒ (ii). Suppose that A is similar to C = ⊕ki=1 Ci , where each Ci is of type (a) or (b). Let i ∈ {1, . . . , k}. If Ci = C(q), for some q(x) ∈ FM [x] such that γ(q(x)) = q(x), then, by Lemma 4(b), Ci is similar to −Ci−1 via an involution Pi . If Ci = C(g) ⊕ C(γ(g)) for some g(x) ∈ FM [x] of degree m, we show that Ci is similar to −Ci−1 via an involution. If Pi :=
0 (−1)m+1 Tm Sm
Tm Sm , 0
then Pi is an involution and, by Lemmas 4(a) and 3(b), Pi−1 Ci Pi = −Ci−1 . If we take Q := ⊕ki=1 Pi , then Q is an involution such that Q−1 CQ = −C −1 . By Lemma 2, A is similar to −A−1 via an involution. This proves that (v) implies (ii). 2 Declaration of competing interest There is no competing interest. Acknowledgement The work of J.R. Salinasan was funded by the ASTHRDP-NSC of the Department of Science and Technology - Science Education Institute. The authors thank the referee for the many helpful comments to improve the paper. References [1] D.Ž. Djoković, Product of two involutions, Arch. Math. 18 (1967) 582–584. [2] E.W. Ellers, Products of two involutory matrices over skewfields, Linear Algebra Appl. 26 (1979) 59–63.
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[3] S. Furtado, Products of two involutions with prescribed eigenvalues and some applications, Linear Algebra Appl. 429 (2008) 1663–1678. [4] T.W. Hungerford, Algebra, Springer-Verlag, New York, 1974. [5] M.J. Wonenburger, Transformations which are products of two involutions, J. Appl. Math. Mech. 16 (1966) 327–338.
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
The product of an involution and a skew-involution Agnes T. Paras ∗ , Jenny R. Salinasan Institute of Mathematics, University of the Philippines, Diliman, Quezon City 1101, Philippines
a r t i c l e
i n f o
Article history: Received 18 September 2019 Accepted 25 September 2019 Available online 30 September 2019 Submitted by V.V. Sergeichuk
a b s t r a c t We give necessary and sufficient conditions for a matrix over a field to be the product of an involution (A2 = I) and a skew-involution (A2 = −I). © 2019 Elsevier Inc. All rights reserved.
MSC: 15A21 15A23 Keywords: Involution Skew-involution Companion matrix Frobenius canonical form
1. Introduction Let Mn (F) denote the set of all n-by-n matrices over a field F . Suppose A ∈ Mn (F ). The matrix A is said to be an involution if A2 = I. Wonenburger proved in 1966 that if the characteristic of F is not equal to 2, then A is a product of two involutions if and only if A is similar to A−1 [5]. A year after, Djoković gave a proof of the preceding result for an arbitrary field [1]. Ellers showed in 1979 that the result fails to hold when F is a * Corresponding author. E-mail addresses: [email protected] (A.T. Paras), [email protected] (J.R. Salinasan). https://doi.org/10.1016/j.laa.2019.09.035 0024-3795/© 2019 Elsevier Inc. All rights reserved.
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A.T. Paras, J.R. Salinasan / Linear Algebra and its Applications 584 (2020) 431–437
noncommutative field [2]. In 2008, Furtado gave necessary and sufficient conditions for A to be a product of two involutions with prescribed eigenvalues [3]. We say that B ∈ Mn (F) is a skew-involution if B 2 = −I. If the characteristic of F is two, then B is a skew-involution if and only if B is an involution. We wish to find necessary and sufficient conditions for A ∈ Mn (F) to be a product of an involution and a skew-involution. Suppose the polynomial x2 + 1 has a root u in F . Let A ∈ Mn (F ) be given. Notice that A is a skew-involution if and only if uA is an involution. Hence, A = ST , where S is an involution and T is a skew-involution if and only if uA = S(uT ) is a product of two involutions. By the theorem of Wonenburger and Djoković, the latter statement is equivalent to uA is similar to (uA)−1 = −uA−1 . Therefore if x2 + 1 has a root in F , then A is a product of an involution and a skew-involution if and only if A is similar to −A−1 . We show that this is also the case when x2 + 1 has no root in F . 2. The Frobenius canonical form of −A−1 Denote by GLn (F) the set of all nonsingular matrices in Mn (F ). Let F [x] be the set of all polynomials in x over F , and denote the minimal polynomial of A ∈ Mn (F ) by mA (x). n Let h(x) be a monic polynomial over F with positive degree n, say h(x) = ai xi . The i=0
Frobenius block or companion matrix of h(x) is defined to be ⎡
0 ··· ⎢ ⎢1 C(h) = ⎢ .. ⎢ . ⎣
⎤ −a0 ⎥ −a1 ⎥ .. ⎥ ⎥ ∈ Mn (F ). . ⎦ 1 −an−1 0
It is known that the minimal and characteristic polynomials of C(h) are both equal to h(x). The Frobenius (or rational) canonical form theorem states that if A ∈ Mn (F ), then A is similar to a direct sum of Frobenius blocks ⊕ki=1 C(pei i ), where each pi (x) ∈ F [x] irreducible and each ei is a positive integer. The direct sum ⊕ki=1 C(pei i ) is unique up to permutation of summands. The polynomials pi (x)ei for i = 1, . . . , k are called the elementary divisors of A ([4], Corollary 4.7). Thus two matrices in Mn (F ) are similar if and only if they have the same elementary divisors. In particular, if x2 + 1 has no root in
0 −I n2 F and T ∈ Mn (F) is a skew-involution, then n is even and T is similar to , I n2 0 since mT (x) = x2 + 1 is irreducible in F [x]. An immediate consequence of the Frobenius canonical form theorem is the following. Corollary 1. Let A ∈ Mn (F) and B, C ∈ Mm (F) be given. If A ⊕ B is similar to A ⊕ C, then B is similar to C.
A.T. Paras, J.R. Salinasan / Linear Algebra and its Applications 584 (2020) 431–437
433
Let A, B ∈ GLn (F) such that A = T −1 BT for some T ∈ GLn (F ). Suppose there is an involution P ∈ GLn (F) such that P −1 AP = −A−1 . If Q := T P T −1 , then Q2 = I and Q−1 BQ = T P −1 T −1 BT P T −1 = T P −1 AP T −1 = T (−A−1 )T −1 = −B −1 . Lemma 2. Suppose A, B ∈ GLn (F) such that A is similar to B. If there exists an involution P ∈ GLn (F) such that P −1 AP = −A−1 , then there exists an involution Q ∈ GLn (F ) such that Q−1 BQ = −B −1 . Suppose R is the Frobenius canonical form of A ∈ GLn (F ). Lemma 2 implies that if we wish to determine whether A is similar to −A−1 via an involution, then it suffices to determine whether R is similar to −R−1 via an involution. Let FM [x] := {g(x) ∈ F [x] | g(x) is monic of positive degree and g(0) = 0}. If g(x) =
n
ai xi ∈ F [x] is of degree n, then g(x) ∈ FM [x] if and only if n ∈ Z+ , an = 1,
i=0
and a0 = 0. Define the degree-preserving function γ : FM [x] → FM [x] by γ(
n
n
r i xi ) =
i=0
i=0
(−1)n−i r0−1 rn−i xi .
Then γ(g(x)) = g(x) if and only if r0 = (−1)n r0−1 and ri = (−1)n−i r0−1 rn−i for 0 < i < n, which is equivalent to r02 = (−1)n and ri = (−1)i r0 rn−i for 0 < i < n. Thus if γ(g(x)) = g(x) and x2 + 1 has no root in F , then n is even and r0 = ±1. We consider other properties of γ in the following lemma. Lemma 3. Let g(x), q(x) ∈ FM [x] of degree n be given. (a) If g(x) =
n i=0
ri xi , then γ(g(x)) = g(x) if and only if r02 = (−1)n and ri =
(−1)i r0 rn−i for 0 < i < n. (b) If γ(q(x)) = g(x), then γ(g(x)) = q(x), that is γ 2 = idFM [x] . (c) The function γ is multiplicative. (d) If g(x) is irreducible, then γ(g(x)) is irreducible. Proof. Let g(x) =
n
ri xi , q(x) =
i=0
n
si xi ∈ FM [x] such that γ(q(x)) = g(x). Then
i=0
n −1 (−1)n−i s−1 and sn−i = 0 sn−i = ri for 0 ≤ i < n, which is equivalent to s0 = (−1) r0 i −1 n −1 n−i −1 (−1) r0 ri for 0 < i < n. This implies s0 = (−1) r0 and si = (−1) r0 rn−i for 0 < i < n, that is q(x) = γ(g(x)). This proves (b).
To show (c), let v(x) =
m i=0
ai xi ∈ FM [x] have degree m. Then
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γ(v(x))γ(g(x)) =(
m j=0
=
j (−1)m−j a−1 0 am−j x )(
m+n
n
k=0
(
(−1)n−k r0−1 rn−k xk )
(−1)m+n−j−k (a0 r0 )−1 am−j rn−k )xi
i=0 j+k=i
=
m+n
(−1)m+n−i (a0 r0 )−1 (
i=0
= γ[
aj rk )xi
j+k=m+n−i
m+n
(
aj rk )xi ] = γ(v(x)g(x)).
i=0 j+k=i
Finally, suppose that γ(g(x)) is reducible, i.e., there exist non-constant f (x), h(x) ∈ F [x] such that γ(g(x)) = f (x)h(x). Note that f (x) and h(x) can be chosen to be monic; if not, take f1 (x) = r−1 f (x) and h1 (x) = s−1 h(x), where r and s are the leading coefficients of f (x) and h(x), respectively, so that γ(g(x)) = f1 (x)h1 (x). Moreover, the constant term of γ(g(x)) is nonzero, so the constant terms of f (x) and h(x) are nonzero. Hence, we can assume without loss of generality that f (x), h(x) ∈ FM [x]. By (b) and (c), g(x) = γ(γ(g(x))) = γ(f (x)h(x)) = γ(f (x))γ(h(x)), that is g(x) is reducible. This proves the contrapositive of (d). 2 k If g(x) = i=1 pi (x)li is a factorization of g(x) ∈ FM [x] into monic, irreducible k polynomials in F [x], then, by Lemma 3, γ(g(x)) = i=1 γ(pi (x))li is a factorization of γ(g(x)) into monic, irreducible polynomials in F [x].
Let Tn = diag −1, 1, −1, 1, . . . , (−1)n−1 , (−1)n ∈ Mn (F ). Denote by Sn = [sij ] the n-by-n backward identity matrix, that is sij = 1, if i + j = n + 1, and sij = 0, otherwise. Observe that (Tn Sn )2 = (−1)n+1 I, that is Tn Sn is an involution when n is odd; Tn Sn is a skew-involution when n is even. It can be readily verified that if g(x) ∈ FM [x] is of degree n, then C(γ(g))Tn Sn C(g) = −Tn Sn , that is −C(g)−1 is similar to C(γ(g)) via Tn Sn . Lemma 4. Let g(x) ∈ FM [x] of degree n be given. (a) (Tn Sn )−1 C(γ(g))Tn Sn = −C(g)−1 . (b) If γ(g(x)) = g(x), then there exists an involution P ∈ GLn (F ) such that P −1 C(g)P = −C(g)−1 . Proof. Suppose g(x) =
n
ri xi ∈ FM [x] is of degree n and γ(g(x)) = g(x). By
i=0
Lemma 3(a), we have r02 = (−1)n and ri = (−1)i r0 rn−i , for 0 < i < n. Define
A.T. Paras, J.R. Salinasan / Linear Algebra and its Applications 584 (2020) 431–437
⎡
1 ⎢ (−1)n r0 r1 ⎢ ⎢ (−1)n r r 0 2 ⎢ ⎢ . ⎢ .. P =⎢ ⎢ n ⎢ (−1) r0 rn−3 ⎢ ⎣ (−1)n r0 rn−2 (−1)n r0 rn−1
0 0 0
0 0 0
0 0 (−1)n r0
0 (−1)n−1 r0 0
··· ··· ···
0 0 0
··· ··· ···
.. 0 0 0
0 0 (−1)3 r0 . 0 0 0
435
⎤ 0 (−1)2 r0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ 0 ⎥ ⎦ 0 0
Then P is an involution, ⎡ ⎢ ⎢ ⎢ P C(g) = ⎢ ⎢ ⎢ ⎣
−r0
0 r0 ..
.
(−1)n−1 r0 n
(−1) r0
⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
0
and [P C(g)]2 = (−1)n+1 r02 I = −I, since r02 = (−1)n and ri = (−1)i r0 rn−i for 0 < i < n. This proves (b). 2 If A ∈ GLn (F) is similar to ⊕ki=1 C(gi ) for some gi (x) ∈ FM [x], then, by Lemma 4(a), −A−1 is similar to ⊕ki=1 C(γ(gi )). 3. Main result We now give necessary and sufficient conditions for A ∈ GLn (F ) to be a product of an involution and a skew-involution. Theorem 5. Let A ∈ GLn (F) be given. The following are equivalent: (i) (ii) (iii) (iv) (v)
A is a product of an involution and a skew-involution. A is similar to −A−1 via an involution. A is similar to −A−1 via a skew-involution. A is similar to −A−1 . The Frobenius canonical form of A is a direct sum of matrices of types (a) C(g) ⊕ C(γ(g)) for some g(x) ∈ FM [x], (b) C(g) for some g(x) ∈ FM [x] such that γ(g(x)) = g(x).
Proof. Let A ∈ GLn (F) be given. (i) ⇐⇒ (ii) ⇐⇒ (iii). If A = ST , where S is an involution and T is a skewinvolution, then S −1 AS = T S = −A−1 , that is A is similar to −A−1 via an involution. If S −1 AS = −A−1 for some involution S, then AS is a skew-involution and (AS)−1 A(AS) = S −1 AS = −A−1 . If K −1 AK = −A−1 for some skew-involution K,
436
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then AK is an involution and A = (AK)(−K) is a product of an involution and a skew-involution. We have shown that (i), (ii), and (iii) are equivalent. (iii) =⇒ (iv). This is immediate. (iv) =⇒ (v). Suppose that A is similar to −A−1 . Let A be similar to ⊕m i=1 C(gi ), where g1 (x), . . . , gm (x) ∈ FM [x] are the elementary divisors of A. By Lemma 4(a), we m have ⊕m i=1 C(gi ) is similar to ⊕i=1 C(γ(gi )). Without loss of generality, let N ∈ {1, . . . , m} be such that γ(gi (x)) = gi (x) if and only if i > N . By Corollary 1, ⊕N i=1 C(gi ) is similar N to ⊕i=1 C(γ(gi )), where γ(gi (x)) = gi (x) for i ≤ N . If i1 ∈ {1, . . . , N }, then gi1 (x) = γ(gj1 (x)) for some j1 ∈ {1, . . . , N } \ {i1 }. By Lemma 3(b), gj1 (x) = γ(gi1 (x)). Set H1 = {1, . . . , N } \{i1 , j1 }. If H1 = ∅, then N = 2 and A is similar to C(gi1 ) ⊕C(γ(gi1 )) ⊕ ⊕m j=N +1 C(gj ), where γ(gj (x)) = gj (x) for all j > N . If H1 = ∅, then, by Corollary 1, ⊕i∈H1 C(gi ) is similar to ⊕i∈H1 C(γ(gi )). We repeat the above argument to conclude N/2 that N is even and A is similar to ⊕i=1 [C(qi ) ⊕ C(γ(qi ))] ⊕ ⊕m j=N +1 C(gj ) for some qi (x), gj (x) ∈ FM [x] such that γ(qi (x)) = qi (x) for all i and γ(gj (x)) = gj (x) for all j. This proves that (iv) implies (v). (v) =⇒ (ii). Suppose that A is similar to C = ⊕ki=1 Ci , where each Ci is of type (a) or (b). Let i ∈ {1, . . . , k}. If Ci = C(q), for some q(x) ∈ FM [x] such that γ(q(x)) = q(x), then, by Lemma 4(b), Ci is similar to −Ci−1 via an involution Pi . If Ci = C(g) ⊕ C(γ(g)) for some g(x) ∈ FM [x] of degree m, we show that Ci is similar to −Ci−1 via an involution. If Pi :=
0 (−1)m+1 Tm Sm
Tm Sm , 0
then Pi is an involution and, by Lemmas 4(a) and 3(b), Pi−1 Ci Pi = −Ci−1 . If we take Q := ⊕ki=1 Pi , then Q is an involution such that Q−1 CQ = −C −1 . By Lemma 2, A is similar to −A−1 via an involution. This proves that (v) implies (ii). 2 Declaration of competing interest There is no competing interest. Acknowledgement The work of J.R. Salinasan was funded by the ASTHRDP-NSC of the Department of Science and Technology - Science Education Institute. The authors thank the referee for the many helpful comments to improve the paper. References [1] D.Ž. Djoković, Product of two involutions, Arch. Math. 18 (1967) 582–584. [2] E.W. Ellers, Products of two involutory matrices over skewfields, Linear Algebra Appl. 26 (1979) 59–63.
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[3] S. Furtado, Products of two involutions with prescribed eigenvalues and some applications, Linear Algebra Appl. 429 (2008) 1663–1678. [4] T.W. Hungerford, Algebra, Springer-Verlag, New York, 1974. [5] M.J. Wonenburger, Transformations which are products of two involutions, J. Appl. Math. Mech. 16 (1966) 327–338.