Matrix Kloosterman sums modulo prime powers

We give optimal bounds for matrix Kloosterman sums modulo prime powers extending earlier work of the ﬁrst two authors on the case of prime moduli. These exponential sums arise in the theory of the horocyclic ﬂow on GL n .


Introduction
The purpose of this paper is to give good upper bounds for the sums with given A, B ∈ Z n×n where for an n×n matrix X we let ψ(X) = e 2πi Tr X .Note that ψ(XY ) = ψ(Y X).
A good upper bound may mean different things, it could be optimal, or somewhat crude but easily usable, and we will provide both.These sums themselves are of independent interest.They arise naturally in certain equi-distribution problems and are natural analogues of the classical Kloosterman sums x (p) e 2πi(ax+bx −1 )/p .
In our earlier paper [6] we dealt with the case k = 1.As usual upper bounds modulo a prime require the heavy machinery of some type of Weil cohomology.For higher prime powers the methods are usually of a very different sort, based on Taylor expansions, and occasionally referred to as the stationary phase [3].One such example is provided by Salie's explicit evaluation of the one-dimensional case in [14].We will prove that such a result holds generically even for n × n matrices, but generic here is much more restricted than being non-zero, or even invertible.
We summarize the main results.Clearly if p divides all matrix entries of A and B then one may clear appropriate powers and either arrive at a trivial sum, or one where one of A or B is different from the zero-matrix.From now on we will assume that this is the case and denote it as gcd(A, B, p) = 1.
First we have the following reductions to a counting argument.Proposition 1.1.Assume gcd(A, B, p) = 1 and that k > 1.
where the sum is over X ∈ GL n (Z/p k Z) satisfying XAX ≡ B mod p l .
2. If k = 2l + 1 then where the sum is over X ∈ GL n (Z/p k Z), XAX ≡ B mod p l , and where The case when A is invertible modulo p is special and can be made more explicit, in complete analogue of Salie's evaluation of the classical one dimensional Kloosterman sum [14].
1. K n (A, B; p k ) = 0 unless the invariant factors, (the Smith normal forms), of A and B agree up to mod p l , where l = [k/2].
In particular if k > 1 and gcd(det A, p) = 1 (ie.A is invertible mod p) then K n (A, B; p k ) = 0 unless gcd(det B, p) = 1 as well.
2. Assume that gcd(det(AB), p) = 1 and that AB is regular semisimple mod p, (i.e.all eigenvalues are different).Then Note that in the regular semisimple case we have square root cancellation.Also note that we do not assume that the eigenvalues of AB are defined over F p .Finally, this condition is generic, its complement is a Zariski closed set.
We now return to the non-generic cases.By Proposition 1.1 in order to bound the sums in 1 we need to bound Note that for any X ∈ GL n (Z/p l Z) and so if N * (A, B; p l ) is not zero, then it equals N * (C, C; p l ) for C = XA, for any X for which XA ≡ BX −1 (mod p l ).
For l = 1 it is possible (see Thm. 1.5 below) to describe N * (C, C; p) explicitly.This allows one to show that for a well defined exponent e = e C we have m n < N * (C, C; p)/p e < M n for some absolute constants m n , M n that depend on n only.The exponent e = e C itself depends on the combinatorial type of the Jordan decomposition of C over an algebraic closure of F p .
As a first step towards this goal we have the following reduction.
Proposition 1.4.For any be the minimal polynomial of C 2 , where the f j (x) ∈ F p [x] are irreducible.Let V j = ker f j (C 2 ) k j and C j = C |V j be the restriction of C to V j .Then we have In case of a primary minimal polynomial the explicit counting formulas depend on the value of where as usual [13] x f (x) is the quadratic residue symbol, defined for an irreducible polynomial f , To state our main result we need to introduce further notation the details of which are presented in Section 3.For any partition λ = [n 1 , ..., n k ] we let N λ be a nilpotent matrix with Jordan blocks of size n 1 , ..., n k .For q = p d let F q be the field with q elements, and Z GL |λ| (Fq) (N λ ) be the centralizer of N λ in the group GL |λ| (F q ) where |λ| If we have partitions µ, ν we let λ = µ + ν be their join.Also the dual partition λ ′ of λ may be defined via the matrix N λ as λ Theorem 1.5.Assume that p = 2. Let C be a m × m matrix and assume that the minimal polynomial of C 2 is of the form f (x) k , where f (x) ∈ F p [x] is irreducible.We let q = p d , d = deg f , and λ be the partition of m/d with dual , where q = p deg f and 2. If = −1 then all d j will be even, and λ = µ + µ for some partition µ.Then we have where we put r j := d j − d j+1 (j = 1, . . ., k where d k+1 := 0), ie.r j is the number of blocks of size j × j (j = 1, . . ., k) in the Jordan normal form of C.
We are now ready to state our main bounds.For a refined statement we need the stable rank of a matrix A defined as rk ∞ A = lim m→∞ rk A m .Theorem 1.6.Assume that p = 2 and l ≥ 1.
2. Assume that r = rk(A mod p) = rk(B mod p) > 0. We have 3. In particular if n > 1 and gcd(A, B, p Remark 1.7.The case p = 2 is special, in view of (14).Assume AX = X −1 B = C mod p.If C = I n , in particular, if AB = I n mod 2, the bound 2 l(n 2 −n) still holds.
We also need the following general bound for the sum S A,B (X; p).
Proposition 1.8.Assume that A, B are such that there exists X so that AX ≡ X −1 B mod p l .Let with S = (AX − X −1 B)/p l and T = AX mod p. Assume that p = 2.We have that where r = rk T , r ∞ = rk ∞ T .In particular, we always have |S A,B (X; p)| ≤ p n 2 −n and under the additional assumption that A is invertible, we have |S A,B (X; p)| ≤ p n 2 /2 .
In view of Proposition 1.1 as a corollary of the above we have the following Theorem 1.9.Assume that n > 1 and the matrices A, B are not both 0 mod p.We then have the following bounds.
2. If k > 1 we have The paper is organized as follows.First in section 2 we prove Proposition 1.1, this then gives the optimal bounds for the generic situation.In the next section we list some facts concerning partitions, the Sylvester equation and multivariable Gauss sums.These will be used in the following sections.First in Section 4 we give upper bounds for the number of solutions of various quadratic equations in matrices modulo a prime.In the last section Section 5 we then prove the estimates in the last three statements above.
While this work was in progress El-Baz, Lee and Strömbergsson [5] independently arrived to quantitavely similar bounds in their work on the equidistribution of rational points on horocycles.While there are some overlaps the main results are different in nature.

Acknowledgements
Erdélyi was supported by the MTA-RI Lendület "Momentum" Analytic Number Theory and Representation Theory Research Group, and by NKFIH Research Grants FK-127906 and K-135885.
Tóth was supported by by the Rényi Institute Lendület Automorphic Research Group, by the NKFIH Research Grants K 119528 and K-135885.
Zábrádi was supported by the Rényi Institute Lendület Automorphic Research Group, by the MTA-RI Lendület "Momentum" Analytic Number Theory and Representation Theory Research Group, and by the NKFIH Research Grants FK-127906 and K-135885.
We thank the referee for a careful reading and helpful comments.We are also grateful to the authors of the paper [5] who provided valuable suggestions for improvements in the presentation of the paper.We also thank them for many stimulating conversations during a Heilbronn research workshop on Effective equidistribution in homogeneous dynamics, that they organized at the University of Bristol.

Reduction to counting
In this section we prove Proposition 1.1 and its corollaries.As in the statement we need to deal with the case of even and odd exponents separately.

The case p 2l
Let k = 2l.For any unit As noted above, ψ(BUX −1 ) = ψ(X −1 BU), and so the inner sum vanishes, unless AX ≡ X −1 B mod p l .This proves the first claim in Proposition 1.1.

The case p 2l+1
Let k = 2l + 1.We again use the subgroup H defined above which in this case consists of matrices U = I + U 1 p l + U 2 p 2l where U 1 (resp.U 2 ) runs on (Z/p l Z) n×n (resp.on (Z/pZ) n×n ), with inverse where U = I + U 1 p l + U 2 p 2l is such that U 1 will run mod p l and U 2 will run mod p.Now fix X ∈ GL n (Z/p k Z), and consider where Note that S 2 = 0 unless AX ≡ X −1 B mod p in which case S 2 = p n 2 .S 1 is a Gauss sum in matrices, albeit a very special one.By the condition from S 2 we have that AX − X −1 B = pM for some integral matrix M, then we have where T ≡ X −1 Bp l−1 ≡ AXp l−1 mod p l .This gives the claim when l = 1.For l > 1 note that in view of pT ≡ 0 mod p l we have for any V that and so that S 1 (X) = ψ(MV /p l−1 )S 1 (X).A suitable choice of V shows that S 1 (X) = 0 unless M ≡ 0 mod p l−1 .In the original matrices A, B this is equivalent to AX ≡ X −1 B mod p l in which case S 1 = p (l−1)n 2 S A,B (X; p).This gives the second claim of Proposition 1.1.

2.3
The regular semisimple case.
regular semisimple then all the eigenvalues of Y are different and no two of them sum to 0. This is exactly the condition (see subsection 3.3) to modify Y by adding a suitable p l Z in such a way that Y 2 ≡ AB hold mod p k as well.In this case and the claim is an easy corollary of the calculations done in previous two subsections and the regularity of Y 2 .
3 Technical background

Partitions
A partition of an integer n is an ordered set We will write n = |λ|.If λ and µ are two partitions, λ + µ is the partition obtained by taking the parts of λ and µ together (and ordering them).We denote by [n] the partition with one part n.In general if a number j appears r j times in λ, the sequence [..., j, ..., j, ...] will be replaced by [..., j r j , ...], so for example the partition with n parts all equal to 1 is written as [1 n ].Given a partition λ its associated Young (Ferrer) diagram has r rows with n 1 , n 2 , ...n r boxes in each row.For example for λ = [4, 3, 1] we have the diagram The transpose of the diagram of λ is also a Young diagram of a partition λ ′ called the conjugate or dual partition to λ which may be described as follows.Let r i = r i (λ) be the number of parts of λ which are equal to i ≥ 1 and which has the diagram At first assume p = 2 and let N be the nilpotent transformation of an F q vector space V of dimension n.Then V becomes an F q [T ]-module where T acts via N, T v = Nv.Such modules are isomorphic to the module which is unique by the structure theorem of finitely generated modules over principal ideal domains.To show the partition λ associated to N we will use the notation N = N λ .Note that the the dual partition λ ′ arises from considering d i = dim(ker(N i )) − dim(ker(N i−1 )).To see this assume we switch to the matrix point of view and assume that N ∈ M n (F q ) is a nilpotent matrix over F q with r j blocks of size j × j (j = 1, . . ., n 1 ) in the Jordan normal form of N. Let d i = j≥i r j .as above.Then it is easy to see that d 1 is the number blocks, which also equals dim ker N. The claim then follows from an easy inductive argument.One can alternatively define Finally we will need the order of the centralizer of unipotent elements in GL n (F q ).Note that the centralizer of the unipotent I + N is the same as the centralizer of the nilpotent transformation N.

Sylvester's equation
Assume that A is m × m, B is n × n and X and C are m × n matrices.The matrix equation [17], has a rich literature over the real, or complex fields in view of the important role it plays in various applications.(See.e.g.[4].)There are two important questions here, existence of solutions, and a description of all solutions {X | AX − XB = C}.
For our task of estimating K n (A, B; p k ) we will concentrate on estimating the number of solutions.If the field of coefficients is F q for some p-power q, then the number of solutions is clearly either 0 or q d , where While the bound by mn is trivial, in the case when A = λI n , B = λI m for the same scalar λ, one has d A,B = mn.
Note that we may interpret the equation via linear transformations.To do so let W = F m q , V = F n q viewed as column vectors.Both W and V become F q [T ]-modules via mapping T to A and B respectively.If AX = XB then g(A)X = Xg(B) for any polynomial g ∈ F q [T ] and so X gives rise to a module homomorphism from W to V which we denote hom Fq[T ] (W, V ).
For an irreducible polynomial f ∈ F q [T ] let and similarly for which implies hom the sum over f ∈ F q [T ] irreducible.Since the problem is linear, we may go to a finite field extension if needed and then assume that the eigenvalues of A, B are in F q .For f = T − λ and e ≥ 1 we let As an immediate corollary of the trivial bound (9 we get the following Note that that for A semisimple the inequality above becomes an equality, showing that the bound is sharp.
The application for us involves the special case when B = −A.If A is invertible the above bound is sufficient, but the nilpotent case needs a more refined version given in the following lemma.
Proof.As above let W = F m q , V = F n q viewed as F q [T ] modules via mapping T to A and B. Then for some partitions (m 1 , ..., m k ), (n 1 , ..., n l ) of m and n respectively (with k, l as defined above).Note that any element X of hom ) is determined by the value X on 1 mod T b and so by (10) This gives Corollary 3.4.The trivial bound (12) can be strengthened to

Generalities on multivariable Gauss sums.
We start with a general setup on where e(z) = e 2πiz .If L = 0 and p = 2, the sum G(Q; p) is easy to evaluate after diagonalizing Q; it is a product of trivial factors and Gauss sums.The case when p = 2 is slightly more involved [8], but still explicit.
For our use in what follows, some of the details are relevant, and so we sketch these.Assume that p = 2 when we have that Q comes from a bilinear form )), which may not be surjective if B is degenerate.Still we have the following dichotomy.Now assume C is invertible.Since X is also invertible XCX = C if and only if Y = CX satisfies Y 2 = C 2 and we will count the number of solutions to this equation under the assumption that C 2 has minimal polynomial f k (x), with f (x) = x irreducible.For this recall that any linear transformation T has a unique multiplicative Jordan-Chevalley decomposition as where In case p = 2, we can immediately infer that Y u = C u from the following Proof.For simplicity we use the simple property that for any unipotent element Z in GL n (F p ) we have Z p r = I for some r.Since p is odd Therefore estimating N * (A, B; p) is reduced to estimating Our last observation is now the following Lemma 4.2.Put V := F n p and assume that the minimal polynomial of is irreducible and let q = p deg f .Then V has the structure of an F q -vectorspace such that all C, X, Y, Y s , Y u : V → V as above are F q -linear for any invertible solution X of the equation s is an F q -scalar multiple of the identity.
Proof.We assume that f (x) = x otherwise the claim is trivial.Note that F p [x]/(f ) is isomorphic to the field F q with q = p deg f elements.We choose such an isomorphism and let α denote the image of x in F q , so that ] is a semisimple element, it may be identified with α ∈ F * q .Therefore the action of F p [C 2 s ] on V gives an F q -linear structure and since X, Y, Y s , Y u all commute with C 2 s they may be viewed as an F q -linear transformation.Finally, we have

The proof of Theorem 1.5 in the invertible cases
Let V := F m p and C : V → V be an invertible F p -linear map such that we have m C 2 (x) = f k (x) for the minimal polynomial of C 2 with some irreducible polynomial f (x) ∈ F p [x].By Lemma 4.2 we even have an F q -linear structure on V (with q := p deg f ) such that both C and any solution X to the equation CX = X −1 C are F q -linear.Further, C has the Jordan-Chevalley decomposition C = C u C s = C s C u with C u unipotent and C s semisimple.At first assume p = 2 and let N be the nilpotent transformation C u − I. Then V becomes an F q [T ]-module where T acts via N, T v = Nv.Such modules are isomorphic to the module for some partition λ = [n 1 , .., n k ], n 1 + .. + n k = n which is unique by the structure theorem of finitely generated modules over PIDs.To show the partition λ associated to N we will use the notation N = N λ .
By Lemmas 4.1 and 4.2 it is enough to count elements in the set ie. we have N * (A, B; p) = n(C, C; p) = #R α (λ) where α ∈ F * q denotes the unique eigenvalue of C 2 as an F q -linear.
We start with the case when = 1 when we have α = β 2 for some β ∈ F * q .We have the following Proof.Since p = 2, Y 2 = αI implies Y is semisimple whence V is the direct sum of the two eigenspaces of Y .Moreover, these eigenspaces are N-invariant, ie.they are F q [T ]-submodules.Conversely, given such a decomposition V = U + ⊕ U − , we have Y 2 U = αI and Y U commutes with N. By the theorem of elementary divisors, for any decomposition V = U + ⊕ U − of the F q [T ]-module V , λ is the sum of the multisets µ and ν where µ (resp.ν) is the partition of dim U + (resp. of dim U − ) corresponding to the restriction of N to U + (resp.to U − ).Therefore we may write S(λ) as the union of where µ runs over the multisets included in λ and ν = λ − µ is the difference.Proof.Assume we are given two decompositions Taking the direct sum of these two isomorphisms we obtain an automorphism g : ]-linear it means that g lies in Z GL |λ| (Fq) (N) when viewed as an F q -linear transformation.
Proof.By Lemma 4.3 we obtain #R α (λ) = #S(λ) = λ=µ+ν #S(λ, µ, ν).The statement follows from Lemma 4.4 noting that the stabilizer of a given decomposition = −1, so we have α = β 2 for some β ∈ F * q 2 \ F * q .Put σ for the nontrivial element in Gal(F q 2 /F q ).Then ϕ = σ ⊗ I : . By the structure theorem for finitely generated modules over the PID F q 2 [T ], we must have U ∼ = U ′ as F q 2 [T ]-modules, as well.Taking such an isomorphism S : U → U ′ we also define S(ϕ(u)) := ϕ(S(u)) on ϕ(U) giving rise to an F q 2 [T ]-linear automorphism S : Moreover, S descends to a map S : V → V (such that S = 1 ⊗ S) since it commutes with ϕ.Finally, S satisfies SY S −1 = Y ′ and lies in the centralizer of N λ as it is F q [T ]-linear.
Proposition 4.7.Assume α / ∈ (F * q ) 2 .Then we have λ = µ + µ for some partition µ and Proof.By Lemma 4.6 R α (λ) is the conjugacy class of C s in Z GL |λ| (Fq) (N λ ).A moments thought shows that we may define an F q 2 -linear structure on V where C s acts via multiplication by β where the F q 2linear maps are exactly those which are F q -linear and commute with C β .In particular, the centralizer of This leads to formula (5).
Corollary 4.8.Assume that C has minimal polynomial f (x) k where f (x) = x is irreducible and p = 2.
Proof.Since n(C, C, p) is the number of square roots of C 2 s commuting with C u , the case C u = I gives an upper bound for the number of solutions in general.So we may assume N = 0. Put n 1 := dim Fq V = n deg f .Then in the split case we compute (q n 1 − 1) . . .(q n 1 − q n 1 −1 ) (q j − 1) . . .(q j − q j−1 )(q n 1 −j − 1) . . .(q n 1 −j − q n 1 −j−1 ) = On the other hand, we have with constant c(q) = ∞ j=1 (1 − 1/q j ) that clearly satisfies lim q→∞ c(q) = 1.Finally, assume p = 2. Since the 2-Frobenius is bijective on finite fields of characteristic 2, C 2 s has a unique square root Y s = C s .So we need to count the square roots of the unipotent matrix C 2 u or equivalently the square roots of the nilpotent matrix C 2 u + I = (C u + I) 2 .Lemma 4.9.Assume that q is a power of 2. For any integer n > 0 we have the identification as F q [T 2 ]-modules.
Proof.This amounts to the fact that the square of a nilpotent Jordan block of size n splits into two blocks of size ⌊ n 2 ⌋ and ⌈ n 2 ⌉.Proposition 4.10.Assume q is a power of 2. Then the number of solutions of the matrix equation where µ = [m 1 , . . ., m k ] runs on the set of partitions such that Proof.By Lemma 4.9 N 2 µ is similar to N 2 λ if and only if So for each such µ we are reduced to determine the cardinality of the fiber at N 2 λ of the map ker(C) ∩ Im(C j ), so the possible values of X t v is exactly w + (ker(C) ∩ Im(C j )) which has cardinality #(ker(C) ∩ Im(C j )) = p r j+1 +•••+r k .Finally, we let v run on the lift of a basis of the quotient space (ker(C t ) ∩ Im(C j ))/(ker(C t−1 ) + Im(C j+1 )) for any j = k − t, k − t − 1, . . ., 1, 0 (noting Im(C k−t+1 ) ⊆ ker(C t−1 )) we deduce that the number of extensions of X t−1 to a map X t : ker(C t ) → ker(C t ) satisfying (i) and (ii) t is as we have dim Fp (ker(C t ) ∩ Im(C j ))/(ker(C t−1 ) + Im(C j+1 )) = r j+t .
The statement follows from the above lemma by taking t = k: the number of solutions of XCX = C in invertible X equals as claimed.Proof.Using Proposition 4.12 we compute The proofs of the bounds 5.1 The equation AX ≡ X −1 B modulo prime powers Assume that A, B ∈ M n (Z).We are interested in estimating the size of the affine variety V A,B (p l ) where We will collect elements of V A,B (p l+1 ) according to their image in V A,B (p l ).The final push down to l = 1 will play a special role and we let A,B (p l ) be given.Then all X ∈ V A,B (p l+1 ) such that X ≡ X 0 mod p l may be written as X = X 0 (I + p l Y ), for some Y mod p.The goal is to bound Y for which (19) also holds mod p l+1 .This leads to Note that the equation above might have no solution, or exactly as many solution as for which we may apply Proposition 3.3 and its corollary.This gives Proof.This is merely a restatement of Lemma 3.3 and (12).First note that dim ker C = n − r and dim ker C n = n − r ∞ .
To simplify the contribution of the non-zero eigenvalues in (12) use that 2d and that for a 1 , ..., a k positive integers, a 2 1 + ... + a 2 k ≤ (a 1 + ... + a k ) 2 .This estimate is wasteful since the solution set could be empty.However, this will suffice for us.
The proof of Theorem 1.6.When l = 1 the bound for N * (C, C; p) follows from Theorem 1.5 together with exact formulae in Proposition 3.1 which gave Corollaries 4.8 and 4.14.
To see this, note that we may decompose C (over the ground field F p ) as a block matrix with one block invertible of size r ∞ and one block nilpotent of size n − r ∞ .
For the invertible part we have the upper bound 2 r∞ p r 2 ∞ /2 using Corollary 4.8 for each irreducible factor = X of the minimal polynomial of C 2 , noting that there at most r ∞ such factors, and applying a where e(l, n, r, r ∞ ) = (n − r) 2 + (r − r ∞ ) 2 + r 2 ∞ /2 + (l − 1) (n − r)(n − r ∞ ) + r 2 ∞ /2 .In order to prove the second statement, note that unless N * (A, B; p l ) = 0, we find a common value C := AX 0 = X −1 0 B such that N * (A, B; p l ) = N * (C, C; p l ) and put r ∞ := r ∞ (C).Moreover, from Lemma 4.1 the value of r ∞ is the same for any of the C-s that arise.So we are bound to estimate e(l, n, r, r ∞ ).
To simplify the exponent assume first that n/2 ≤ r ≤ n.A calculation shows that the maximum of the function This establishes both bounds in (7).
Finally to prove the universal bound N * (A, B; p l ) ≤ 2 n p l(n 2 −n) note that it holds trivially for l ≥ 2, since n 2 − n ≥ n 2 /2.

Gauss sums of matrices
There are various ways exponential sums with quadratic functions of the entries of an n × n matrix arise.For example in the theory of Siegel modular forms Q(X) = Tr X t AX, and the associated Gauss sums play an important role see e.g [18].These have a very different flavor than ours, as the tensor properties allow one to diagonalize A, which immediately yields a diagonalization of the quadratic form Q(x 11 , x 12 , . . ., x nn ).This approach is not directly applicable to our situation since we have Q(X) = Tr T X 2 for some matrix T .While this case appeared in the literature, see e.g.[7] our treatment is based directly on Proposition 3.5 and its corollary 3.6.
The proof of Proposition 1.

( a )
If k = 2l then K n (A, B; p k ) = p kn 2 /2 Y ψ(2Y /p k ) where the sum is over Y ∈ GL n (Z/pZ), Y 2 ≡ AB mod p k .(b) If k = 2l + 1 then K n (A, B; p k ) = ζp kn 2 /2 Y ψ(2Y /p k )where the sum is over Y ∈ GL n (Z/pZ), Y 2 ≡ AB mod p k , and where ζ is a p-th root of unity.(c)In particular we have |K n (A, B; p k )| ≤ 2 n p kn 2 /2 .
with a quadratic form Q, and a linear form L on V and define G(F ; p) = x∈V e(F (x)/p)

Remark 4 . 15 .k j=1 d 2 j
For fixed n and p → ∞ the above upper estimate p is in fact the order of magnitude of n(C, C, p):

8 .
We have to estimate the sumS A,B (X; p) = U mod p ψ((SU + T U 2 )/p),where X is such that AX ≡ X −1 B mod p l , and where S = (AX − X −1 B)/p l and T = AX mod p.This is clearly a general Gauss sum.To apply Corollary 3.6 let B(U, Y ) = Q(U + Y ) − Q(U) − Q(Y ) =Tr((T Y + Y T )U) be the associated bilinear form.