A New Invariant Under Congruence of Nonsingular Matrices

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2 Invariance and other properties of σ

First of all, let us show that σ is an invariant under congruence of nonsingular matrices. In fact, for any nonsingular matrix X,

− σ(tXAX) = T r(t(tXAX)(tXAX) 1) − − − = T r((tX tAX)(X 1A 1(tX) 1)) − − = T r(tX(tAA 1)(tX) 1) = σ(A).

As an immediate corollary, any polynomial in σ(A) over K is also an invariant under congruence of nonsingular matrices. Moreover, any rational function in σ(A) over K is an invariant under congruence of nonsingular matrices for which its denominator does not vanish. Next, let us describe some other properties of σ. Proposition 1. For any n n nonsingular matrix A over K, the following equalities hold. × (1) σ(tA)= σ(A). (2) σ(A−1)= σ(A). (3) σ(adj(A)) = σ(A), where adj(A) is the adjugate of A. (4) σ(cA)= σ(A) for any c K 0 . ∈ \{ } (5) σ(A)= n if A is symmetric. Proof. (1) − σ(tA) = T r(t(tA) (tA) 1) − = T r(A t(A 1)) − = T r(A 1(tA)) = σ(A)

(2) − − − − σ(A 1) = T r(t(A 1) (A 1) 1) − = T r(t(A 1) A) = σ(A)

2 (3)

− σ(adj(A)) = T r(t(adj(A)) (adj(A)) 1) − = T r(adj(tA) (adj(A)) 1) − − = T r(( tA (tA) 1) ( A 1A)) | |− | | = T r((tA) 1 A) − = T r(t(A 1) A) = σ(A)

(4) For any c K 0 , ∈ \{ } − σ(cA) = T r(t(cA) (cA) 1) − − = T r(c tAc 1A 1) = σ(A)

(5) If tA = A, then

− σ(A) = T r(AA 1) = T r(E) = n

3 Background on σ

Let us briefly describe the background that led to the discovery of σ. The authors of the present note with Sin-Ei Takahasi and Makoto Tsukada tried to classify three-dimensional zeropotent algebras up to isomorphism, where a zeropotent algebra is a nonassociative algebra in which the square of any element is zero, see [1, 3]. We there expressed an algebra by its structure constants. For a three-dimensional zeropotent algebra A, let e1,e2,e3 be a linear base of A. By zeropotency of A it suffices to consider{ the structure} constants for e2e3,e3e1,e1e2, namely, the algebra can be identified with the 3 3 matrix A t t × such that (e2e3 e3e1 e1e2) = A (e1 e2 e3). We hereafter use the same symbol A both for the matrix and for the algebra. The following proposition gives a criterion for two zeropotent algebras to be isomorphic. Proposition 2 ([1]). Let A and A′ be three-dimensional zeropotent algebras over K. Then, A and A′ are isomorphic if and only if there is a nonsingular matrix X satisfying ′ 1 A = tXAX. X | | In particular, if K is algebraically closed, then A and A′ are isomorphic if and only if there is a nonsingular matrix X satisfying A′ = tXAX, that is, A and A′ are congruent.

3 Our strategy of classiﬁcation was to divide zeropotent algebras into curly algebras and straight algebras. A zeropotent algebra is curly if the product of any two elements lies in the space spanned by these elements, otherwise straight. Let K be an algebraically closed ﬁeld. A straight algebra of rank 3 can be expressed by the canonical form

1 a b A(a,b,c)= 0 1 c 0 0 1 with a,b,c K. During∈ the course of classifying the straight algebras, we encountered the quantity D = a2 + b2 + c2 abc, which seems strange because it is not ho- mogeneous with respect to −a, b, and c. That is, in case where D = 0 and a2 + b2 abc = 0, we showed that A(a,b,c) is congruent to A(√D, 0, 0).6 From this, it turned− 6 out that D is an invariant under congruence of some A(a,b,c)- type matrices. Inspired by this fact, we got interested in elucidating the origin of D. From D, later, by backward reasoning from the canonical form to a general matrix using computer, the second author derived a more general ho- 1 2 2 mogeneous quantity a a22 +3a12a23a31 a21a23a31 + a22a + a11(a23 A { 13 − 31 − 2 | | a32) a12a31a32 a21a31a32 a13(a21a23 +2a22a31 3a21a32 +a12(a23 +a32))+ 2 − − 2 − − a12a33 2a12a21a33 + a21a33 , for a 3 3 matrix A = (aij ) of rank 3. Let us denote− this quantity by κ(A).} By computer× we confirmed that κ is certainly an invariant under congruence of general nonsingular matrices, but could not prove it mathematically. Since the original expression of κ(A) was awkward and com- plicated, by an enormous amount of transformations of trial and error, the first 1 author obtained the more beautiful form 3 (aij aji), where aij denotes − A X | | i,j f f the (i, j) cofactor, that is, the product of ( 1)i+j and the (i, j) minor. Denoting 1 − 1 (a a ) by σ(A), by further reasoning he had σ(A)= T r(A adj(tA)), A X ij ji A | | i,j f | | and finally by using tA(adj(tA)) = tA E, reached the simple form T r(tAA−1) as initially defined and then κ(A)=3| | T r(tAA−1). Through this expression we can now mathematically prove that−κ is an invariant under congruence of nonsingular matrices. Beyond the topic of three-dimensional zeropotent algebras, we found that in general, for any integer n 1, σ is an invariant under congruence of n n nonsingular matrices over a general≥ field. ×

References

[1] Y. Kobayashi, K. Shirayanagi, S.-E. Takahasi and M. Tsukada, Classiﬁca- tion of three-dimensional zeropotent algebras over an algebraically closed ﬁeld, Comm. Algebra, 45 12 (2017), 5037–5052.

4 [2] C. Procesi, The invariant theory of n n matrices, Adv. Math., 19 (1976), 306–381. × [3] K. Shirayanagi, S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classiﬁca- tion of three-dimensional zeropotent algebras over the real number ﬁeld, Comm. Algebra, 46 11 (2018), 4663–4681.

K. Shirayanagi and Y. Kobayashi Department of Information Science, Toho University, Miyama 2-2-1 Funabashi, Chiba 274-8510, Japan [email protected] and [email protected]