Below we give an estimate for the probability of finding a proper submodule in the situ-ation where the algebra A satisfies conditions (5.1) and (5.2). Actually we show that the commutator [x, i(x)yi(x)] has a positive chance for being a nonzero element of Rad(A).
Lemma 5.3. Assume that the finite dimensional F-algebra A with identity satisfies con-ditions (5.1) and (5.2). Let ι be an idempotent ofS. Then
(a) [ιEι, ιAι] =ι[E, A]ι, (b) S(ι[E, A]ι)S= [E, A], and
(c) (0)⊂[ιEι, ιAι]⊆Rad(A).
Proof. First we note that since ι commutes with E, ιbι = ιb = bι for every b ∈ E and hence ιEι =ιE = Eι. Part (a) is immediate from the following equalities which hold for every b ∈E and a∈A.
ιb·ιaι−ιaι·ιb=ιbι·aι−ιa·ιιb=ιb·aι−ιa·bι=ι(ba−ab)ι.
To prove part (b), let s, s0 ∈S, b∈E, a∈A. Then
sι[b, a]ιs0 =sιbaιs0−sιabιs0 =bsιaιs0−sιaιs0b= [b, sιaιs0],
where the second equality holds because b commutes with the elements ι, s, s0 ∈ S. From this we infer thatSι[E, A]ιS = [E, SιAιS]. It remains to establish the equalitySιAιS =A.
To this end observe thatSιS is a nonzero ideal in the simple algebra S, thereforeSιS =S.
Hence SιAιS = SιSASιS = SAS = A. (The first and the last equalities are obvious because S contains 1A.)
Part (c) follows from (a) and (b) and the fact that E is central modulo Rad(A).
After these preparations we are ready to give a lower bound on the probability of success of the algorithm.
Proposition 5.4. Assume that the matrix algebra A ≤ Md(F) satisfies conditions (5.1) and(5.2). Then the proportion of the triples (x, y, v)∈A×A×Fd for which the algorithm described in the preceding section finds a proper submodule is at least 0.08.
Proof. Assume thatp(t) is an unrepeated irreducible factor of the characteristic polynomial of x+ Rad(A) on En. Then the degree of p(t) is the dimension (over F) of the kernel of p(x+ Rad(A)). This subspace is obviously a Z(A/Rad(A))-submodule of En, and hence the degree ofp(t) is at leaste= dimF Z(A/Rad(A)) = dimF E. Assume that the degree of p(t) is exactlye. By Lemma 5.1, such a factor does exist for at least 21.4% of the elements x∈ A. Furthermore, all the factors of this kind are characterized as the minimum degree factors amongst the factors of minimal multiplicity of the characteristic polynomial of x on the whole module M.
Referring to the homomorphism F[t]/(c(t)) → A induced by x 7→ x, it is immediate thatι =i(x) is an idempotent. Letx=x+ Rad(A) andι =ι+ Rad(A). Furthermore, the characteristic polynomial of ιx onEn is p(t)x(n−1)e. It follows that ιx and ι have ranke, thereforeιis a primitive idempotent ofA/Rad(A). Henceι(A/Rad(A))ι=ιZ(A/Rad(A))ι.
In particular, ιx ∈ιZ(A/Rad(A)). On the other hand, the minimum polynomial of ιx on ιZ(A/Rad(A)) is of degree e, thereforeιx generates the whole ιZ(A/Rad(A)).
Now [x, ιyι] is a nonzero element of Rad(A) for at least 1− |E|1 ≥ 34 of the elements y∈A, see Lemma 5.5 below, and let us assume in the following that this is the case. Then [x, ιyι] is a nontrivialF-linear transformation and hence the kernel has codimension at least 1. Therefore for at least 1−|F1| ≥ 12 of the elements v ∈M the vector [x, ιyι]v is a nonzero element of the proper submodule Rad(A)M = Rad(M). Putting the bounds together, the algorithm finds a proper submodule with probability at least 0.214· 34 · 12 >0.08.
The proposed method, complemented with the Holt-Rees approach gives an algorithm of Las Vegas type for every case.
We now give the promised proof of the statement used above.
Lemma 5.5. Assume that the finite dimensional F-algebra A with identity satisfies condi-tions (5.1) and (5.2). Assume further that x is an element of A and ι is an idempotent of the subalgebra of A generated by x and 1A such that the subalgebra of A/RadA generated by ιx+Rad(A) is (ι+Rad(A))Z(A/Rad(A)). Then [x, ιAι]⊆Rad(A) and [x, ιyι]6= 0 for at least 1−|E|1 of the elements y∈A.
Proof. Let Ax denote the subalgebra of A generated by ιx. We first note that ι is the identity element of Ax. Indeed, ιa=aι =a holds for every element a∈Ax. On the other hand, it is straightforward to see that A0x =Ax+F ι is a subalgebra andAx is an ideal of A0x. By the assumption
(A0x+ Rad(A))/Rad(A) = (Ax+ Rad(A))/Rad(A)∼=Z(A/Rad(A)),
thusA0x is a local algebra andAxis not a nilpotent ideal. But since in a local algebra every proper ideal is contained in the radical,Ax=A0x, establishing the containment ι ∈Ax.
We are now going to replace S and E with appropriate conjugates in order to achieve the situation where ι ∈ S and ιE is a subalgebra of Ax. By the Wedderburn–Malcev principal theorem, Ax =Sx+ Rad(Ax), where Sx is a semisimple subalgebra ofAx. Since every maximal semisimple subalgebra of A is a conjugate of S by an inner automorphism (cf. [76]), there exists a unit a ∈A such that Sa =a−1Sa ≥Sx. Because conditions (5.1) and (5.2) are invariant under automorphisms, we may replace S with Sa and E with Ea, or, equivalently, assume that Sx ≤S. Note that ι is just the identity element ofSx.
By the assumption
(Ax+ Rad(A))/Rad(A) = (ι+ Rad(A))Z(A/Rad(A))∼=E,
is a simple algebra, therefore Rad(Ax+ Rad(A)) = Rad(A). On the other hand, Rad(Ax) + Rad(A) is obviously a nilpotent ideal of Ax+ Rad(A). It follows that Rad(Ax)≤Rad(A), Ax+ Rad(A) = Sx+ Rad(A) and Sx =ιE.
Observe that, since the idempotent ι commutes withx, for every y∈A we have [x, ιyι] =xιyι−ιyιx=xιιyι−ιyιιx=ιxιyι−ιyιιx= [ιx, ιyι].
The equalityιE =Sx and the preceding lemma give [Sx, ιAι]⊆Rad(A). Since Sx ≤Ax ≤ Sx+ Rad(A), we have
[Ax, ιAι]⊆[Sx, ιAι] + Rad(A)⊆Rad(A).
The first inclusion of the formula above holds because Rad(A) is a two-sided ideal and hence [ιAι,Rad(A)] ⊆ Rad(A). In particular, [x, ιAι] = [ιx, ιAι] ⊆ Rad(A). So we have proved the first part of the statement.
In order to see the second part, notice that, since Sx ≤Ax, CιAι(x) = CιAι(Ax)≤CιAι(Sx)< ιAι.
The latter inclusion is strict because not the whole ιAι commutes with Sx = ιE by Lemma 5.3. Obviously, CιAι(Sx) is an Sx-submodule of ιAι (multiplication by elements fromSx from the left hand side). The set of elementsysuch that [x, ιyι] = 0 is theF-linear subspace (1A−ι)A+A(1A−ι) + CιAι(x). By the preceding argument the codimension of this subspace is at least dimF S = dimFE, whence the second part of the assertion follows.
Chapter 6
Computing the radical of matrix algebras over finite fields
In this chapter, based on the paper [54], we discuss randomized algorithms which compute algebra generators of a Wedderburn complement as well as ideal generators of the radical of a matrix algebra over a finite field given by algebra generators. The cost of the algorithms is comparable to that of a polylogarithmic number of matrix multiplications.
Concerning fast randomized computations in matrix algebras over finite fields given by generators, the first results are due to Eberly and Giesbrecht [30, 31]. They presented randomized algorithms for determining the structure of semisimple matrix algebras over finite fields using a few (i.e. (logn)O(1)) matrix multiplications. These algorithms are nearly optimal, one cannot expect substantially faster methods. The most important result of [30, 31] is valid for arbitrary matrix algebras over finite fields: they can efficiently find a complete system of pairwise orthogonal primitive idempotents.
In Chapter 4 we described a randomized algorithm for computing the radical of matrix algebras over a wide range of ground fields. The number of matrix multiplications per-formed by the method presented therein is roughlyO(n4) if we ignore the cost of producing random algebra elements and that of computing squarefree part of polynomials. In this chapter, based on the algorithm of Eberly and Giesbrecht, we give an improvement on this result in the special case where the ground field is finite. Furthermore, we also compute a subalgebra which is isomorphic to the radical-free part.
We fix some notation used throughout this chapter. The ground field (which is a finite field) is denoted by F. Following the notation introduced in Section 2.1, we assume that O(MM(n)) operations are sufficient to multiply two n×n matrices over F. We also make the following (very reasonable) assumptions on the functionMM(n): MM(n1) +MM(n2) = O(MM(n1+n2)) andn21MM(n2) = O(MM(n1n2)). Notice that these assumptions hold for the examples O(n3) or O(n2.376) given in Section 2.1.
Let A ≤ Mn(F) be a matrix algebra containing the n by n identity matrix In. We denote by V the vector space of the length n column vectors. We assume that A is given by matrices a1, . . . , am, which, together with the identity matrix In generate A as an F -algebra.
Like the MeatAxe procedure discussed in the preceding chapter, the algorithm of Eberly and Giesbrecht [30, 31] for finding a complete system of pairwise orthogonal primitive idempotents presumes the presence of a method for selecting random elements of A in-dependently according to a (nearly) uniform distribution. The number of arithmetical
operations in F performed by the algorithm is O((MM(n) +n2log|F|+R(A))polylogn), whereR(A) stands for the cost of selecting a single random element ofA. Note that, since there can be as many as n pairwise orthogonal idempotents, writing the output as a list of matrices would not fit within the desired complexity bound. Instead, the output is given with the aid of a matrix of a basis transformation such that the idempotents written in the new basis are diagonal.
The algorithm is of Monte Carlo type, i.e., it may return a false output, although with an error probability which can be made arbitrarily small. (Actually, the only possible error is that not all of the idempotents in the system are primitive.) In the semisimple case Eberly and Giesbrecht also showed how to supplement the algorithm with a randomized correctness test of cost within the same complexity bound. This upgrades the algorithm for a semisimple algebraAto a Las Vegas method, i.e., a randomized algorithm which may report failure (with a small error probability) but never returns a false output.
In this chapter we give positive answers to a part of the questions posed in [31]. By the Wedderburn–Malcev principal theorem there exists a subalgebra S ≤ A such that S∩Rad(A) = (0) and A = S + Rad(A). Furthermore, any pair of such subalgebras are conjugated by an element of the form 1+rwherer ∈Rad(A). We refer to such subalgebras S as Wedderburn complements of A. Obviously, a subalgebra S ≤ A is a Wedderburn complement iff S ∼= A/Rad(A). Also, it is an easy consequence of conjugacy part of the principal theorem that Wedderburn complements are just the maximal semisimple subalgebras of A. Note that to construct a Wedderburn complement of A we do not need to work with the whole algebraA. Indeed, ifA0 is a subalgebra such thatA0+ Rad(A) =A then every Wedderburn complement of A0 is a Wedderburn complement of the whole A.
It will be convenient to introduce the following concept. Let S be a Wedderburn complement ofA. Then the mapσS :A→Awhich is the identity onSand zero on Rad(A) is an algebra epimorphism from A to S. We refer toσS as a Wedderburn projection of A.
Assume that S ≤ Mn(F) is a semisimple matrix algebra containing the identity matrix.
By anabsolute Wedderburn projectiontoSwe mean a map σ:Mn(F)→Mn(F) such that σ restricted toSis the identity map and for any matrix algebraA0 ≤Mn(F) havingS as a Wedderburn complement (i.e.,A0 =S+ Rad(A0)),σrestricted to Rad(A0) is the zero map.
For an arbitrary subalgebraA≤Mn(F), an absolute Wedderburn projection ofAis just an absolute Wedderburn projectionσto some Wedderburn complement ofA. We require that σis given by a procedure which computesσ(a) for an arbitrary matrixa. By thecomplexity of σ we mean the maximum number of arithmetical operations sufficient to compute σ(a) for a ∈ Mn(F). Note that we do not require σ to be linear on the whole Mn(F). The advantage of the concept is that an absolute Wedderburn projection of a sufficiently large subalgebra A0 (such that A0 + Rad(A) = A) is automatically a Wedderburn projection of the whole A.
Our main result is an efficient method for finding an absolute Wedderburn projection of complexity roughly O(MM(n)polylogn).
Theorem 6.1. Assume that matrices e1, . . . , es are given such that e1, . . . , es form a complete system of pairwise orthogonal primitive idempotents of A. Then an abso-lute Wedderburn projection σ of A can be constructed by a Las Vegas algorithm per-forming O(m(MM(n) + n2log|F|)polylogn) operations. The complexity of σ is also O((MM(n) +n2log|F|)polylogn).
Applying σ to the generators a1, . . . , am, it is obvious that σ(a1), . . . , σ(am) (together with the identity matrix) generate the Wedderburn complement σ(A) as an F-algebra.
Also,a1−σ(a1), . . . , am−σ(am) generate Rad(A) as an ideal ofA.
Corollary 6.2. Keeping the assumptions of the theorem, m matrices which generate Rad(A) as an ideal of A as well as m matrices which generate a Wedderburn complement of A as an algebra with identity can be calculated by a Las Vegas algorithm performing O(m(MM(n) +n2log|F|)polylogn) arithmetical operations in F.
Combining with the result of [31], we obtain
Corollary 6.3. Assume that we have an auxiliary method which produces random elements ofAindependently and uniformly at the cost ofO(R(A))operations per each element. Then m matrices which generate Rad(A) as an ideal ofA as well asm matrices which generate a Wedderburn complement ofAas an algebra with identity can be calculated by a Monte Carlo algorithm performingO((m(MM(n) +n2log|F|) +R(A))polylogn)arithmetical operations in F.
It turns out that the map σ returned by the algorithm of Theorem 6.1 (if it succeeds) is always a Wedderburn projection to a semisimple subalgebra of A. Thus, to upgrade the algorithm to a Las Vegas method it is sufficient to test nilpotency of the (right) ideal generated bya1−σ(a1), . . . , am−σ(am). In this direction we have the following result.
Theorem 6.4. Let A ≤ Mn(F) be a matrix algebra given by generators a1, . . . , am. Let b1, . . . , bm0 be further elements of A. Assume that S is a subalgebra of A containing the identity matrix such that S and b1, . . . , bm0 generate A. Suppose further that we have an auxiliary procedure for generating random elements of S uniformly and independently at the cost of O(R(S)) operations per each random element. Then there exists a Las Vegas algorithm performingO(((m+m0)n3+nR(S)) log|F|n)operations which either detects that not all of the matrices bi are in Rad(A), or constructs a chain of A-submodules (0) = V0 ≤ V1 ≤ . . . ≤ Vn = V such that biVj ≤ Vj+1 for every i ∈ {1, . . . , m0} and for every j ∈ {1, . . . , n}.
Note that a divide and conquer method based on iterative application of the MeatAxe procedure gives a composition series of V. No accurate complexity analysis of such an algorithm can be found in the literature. We think that the implementation in the C-MeatAxe package requires Ω(n4) operations for certain nilpotent algebras. A chain of submodules with the properties stated in the theorem together with a complete system of primitive idempotents seem to be applicable to construct a composition chain with O(n3polylogn) operations.
Such a chain of submodules witnesses that all the elements bi are in the radical. Indeed, bi must be upper triangular in terms of a basis of V compatible with the chain. We shall also see in Section 6.3 that for a Wedderburn-complement S constructed by the method of Theorem 6.1, after a preprocessing of cost O(mn3) operations, random elements can be drawn using R(S) = O(n2) operations per each. Thus, by combining Theorem 6.4 with Corollary 6.3 we obtain the following.
Corollary 6.5. Under the assumptions of Corollary 6.3, m matrices which generate Rad(A) as an ideal of A as well as m matrices which generate a Wedderburn complement of A as an algebra with identity can be calculated by a Las Vegas algorithm performing O((m(n3+n2log|F|+R(A)))polylogn) arithmetical operations in F.
In Section 6.1, we give a proof of Theorem 6.1 for local algebras. In Section 6.2, using techniques similar to those given in [31], we describe a reduction to the local case. The proof of Theorem 6.4 is given in Section 6.3.
We remark that one could propose variants of the algorithms given in this chapter which use random elements ofAinstead of the generators. The complexity bounds would involve termsR(A) in place of the multiplicative factorm. We have chosen a presentation in terms of generators because – although these variants might be more efficient in practice – it is not known how to generate random elements of A efficiently in a mathematically rigorous way. Also, we hope that the algorithms presented here can contribute to eliminating the random choice of elements of the whole algebra from the algorithm of [30, 31] for finding a complete system of primitive idempotents.
6.1 Wedderburn complements in local algebras
Recall thatA is a local algebra ifA/Rad(A) is a field. Throughout this section we assume A ≤ Mn(F) is a local algebra. Then a Wedderburn complement S of A is a subfield of Mn(F) (containing the identity matrix).
Let C = CA(S). Then S is a Wedderburn complement of C and S ≤ Z(C). By the conjugacy part of the principal theorem, S is the unique Wedderburn complement of C. We call a matrix b ∈ Mn(F) semisimple if the matrix algebra generated by b is semisimple. Since S is the unique maximal semisimple subalgebra of C, it is just the set of semisimple elements. It follows that every element b ∈ C can be uniquely written in the form b = bs+bn where bs ∈ S and bn ∈ Rad(C), or, equivalently, bs is a semisimple matrix andbn is a nilpotent matrix fromC. Of course, this construction also works for the subalgebra generated by b in place of C. The decomposition b=bs+bn is referred as the Jordan decomposition of b. The matrices bn and bs are called the nilpotent part and the semisimple part of b, respectively. The Jordan decomposition can be computed by with O((MM(n) +nlog|F|)polylogn) operations using the Las Vegas rational Jordan normal form algorithm of Giesbrecht [41]. Thus the map b7→bs is a good Wedderburn projection of C.
We use a simple version of the Fitting decomposition technique of Chapter 4. This gives a well defined subspace of A complementary to C as follows. Since S is a finite field there exists an element a ∈ S which generate S as an F-algebra. Of course, a is a semisimple matrix. The linear map ada :b7→ab−bais a semisimple linear transformation onMn(F). Therefore Mn(F) = ker ada+ im ada, a direct sum as vector spaces. Note that this decomposition is inherited by any subspace of Mn(F) invariant under ada, such as A and Rad(A). Also note that ker ada=CMn(F)(a). HenceA=C+ ada(A), a direct sum as vector spaces. Since A/Rad(A) is commutative, ada(A) is in fact a subspace of Rad(A).
In this chapter Φa denotes the map which is the identity on CMn(F)(a) and zero on im ada. A Las Vegas algorithm for computing Φa(b) with O(MM(n)polylogn) operations is described in Section 4.4.
It is straightforward to check that the composition map σa : b 7→ (Φa(b))s) is zero on Rad(A) and maps S identically onto itself. We obtained the following.
Proposition 6.6. Assume that we have a matrix a such that the matrix algebra S gen-erated by a and the identity matrix is a field. Then the map σa : Mn(F) →Mn(F) given above is an absolute Wedderburn projection of S. The complexity of σa is O((MM(n) + nlog|F|)polylogn).
Assume that the matrices a1, . . . , am generate the local algebra A. It is not difficult to show that if the ground field is sufficiently large then the semisimple part of a random linear combination of a1, . . . , am will generate a Wedderburn complement with high probability.
In the rest of this section we describe an iterative algorithm which works over small fields as well. The method is also based on projecting onto the centralizer and then taking the semisimple part.
We calculate a sequencec1, . . . , cm of semisimple elements ofAsuch that the subalgebra Si ofAgenerated byci satisfiesa1, . . . , ai ∈Si+ Rad(A). Thena=cm will be a semisimple matrix such that for the subalgebra S generated by a we have S+ Rad(A) = A. Since S is semisimple S∩Rad(A) = (0) and henceS is in fact a Wedderburn complement.
We start with c0 = In, the n by n identity matrix. Assume that 0 ≤ i < m and we have already calculated a matrix ci with the desired property. Then the subalgebra Si generated by ci is semisimple and hence Si ∩Rad(A) = (0). Therefore the natural projection A→ A/Rad(A) embeds Si into A/Rad(A), which is a field. We first calculate bi+1 = Φci(ai+1) and take the semisimple part di+1 of bi+1. Then di+1 is a semisimple element of A commuting with Si. Since commutative algebras generated by semisimple matrices are semisimple, the algebra Si+1 generated bySi and di+1 is semisimple.
We claim that for every j ∈ {1, . . . , i+ 1}, aj ∈Si+1+ Rad(A). For j ≤i it is obvious from the inductive hypothesis as Si ≤ Si+1. It remains to establish the containment ai+1 ∈Si+1+ Rad(A). SinceA/Rad(A) is commutative, adciA⊆Rad(A) and hence ai+1− Φci(ai+1) ∈ Rad(A). Thus it is sufficient to show that bi+1 = Φci(ai+1) ∈ Si+1+ Rad(A).
But bi+1 −di+1 is nilpotent. In particular bi+1 − di+1 + Rad(A) is a nilpotent element of A/Rad(A). Since A/Rad(A) is a field, this implies bi+1 −di+1 ∈ Rad(A) and hence bi+1 ∈Si+1+ Rad(A). We have proved the claim.
To finish, we need an element ci+1 ∈Si+1 which generate Si+1 as an F-algebra. As by induction Si is generated by ci, therefore Si+1 is a field generated by ci and di+1, this can be done as described below.
Lemma 6.7. Given matrices c, d ∈ Mn(F) such that the matrix algebra S generated by c and d is a field, a single matrix which generates S can be calculated by a Las Vegas algorithm performing O(M M(n) logn+n2lognlog logn) operations.
Proof. We adopt a version of the method of Giesbrecht [41] proposed for evaluating uni-variate matrix polynomials. Let Sc and Sd be the subfields of S generated by c and d, respectively. Setr= dimF S,rc = dimFSc, and rd = dimFSd. Then ris the least common multiple ofrc andrdand the vectorsdjci (i= 0, . . . , rc−1,j = 0, . . . , r/rc−1) form a basis ofS overF. Choose random coefficientsαij fromF uniformly and independently. Then the linear transformation h =P
i,jαijdjci generates S with probability at least 1/4 (cf. [42] , Theorem 5.2). We compute the matrix of h in terms of a special basis of V = Fn. When it is done we write h in terms of the natural basis at the cost of O(M M(n)) operations.
Assume we already know rc, rd and r. (These dimensions can be computed using the Frobenius normal form ofc and d, which will be needed for other purposes, too.) V =Fn can be considered as a vector space of dimension n/r over S. The probability of that n/r random vectorsv1, . . . , vn/r (chosen uniformly and independently fromV) are linearly independent over S is
Suppose that the vectors v1, . . . , vn/r are linearly independent over S. Then the vectors djcjvl (l = 1, . . . , n/r, i = 0, . . . , rc −1, j = 1, . . . , r/rc −1) form a basis of V. In this basis the (l, i, j)th column of h will be the vector P
i0,j0αi0j0dj0ci0djcivl = djciwl, where wl =P
i,jαijdjcivl. Note that in the basis djvivl the vector wl is just the column vector havingαij in the (l, i, j)th position and zero elsewhere. Thus computingh is equivalent to computing the vectorsdjciwl, for l= 1, . . . , n/r,i= 0, . . . , rc−1,j = 1, . . . , r/rc−1. This can be done as follows.
By (a slightly simplified version of) the algorithm of [41] for Frobenius normal forms, we can find matrices uc, ud∈ GLn(F) such that both u−1c cuc and u−1d dud have at most 2n nonzero entries. The algorithm requires O(M M(n) logn+n2lognlog logn) operations.
The simplification is that, in spite of the general case of the algorithm, we do not need to go over an extension field if F is small. Indeed, if we choose random vectors w1, . . . , wn, the first n/rc (resp. n/rd) of them will form a basis ofV over the field Sc (resp. Sd) with probability at least 1/2.
Having uc and ud at hand, we start with computing u−1c vl. This requires O(M M(n)) operations. Then we compute u−1c civl = (u−1c cuc)i(u−1c vl) for i = 0, . . . , rc −1 iteratively.
This can be done with O(nrcn/r) operations because the cost of multiplying a vector by u−1c cuc isO(n) asu−1c cuc has only at most 2nnonzero entries. Next we multiply these
This can be done with O(nrcn/r) operations because the cost of multiplying a vector by u−1c cuc isO(n) asu−1c cuc has only at most 2nnonzero entries. Next we multiply these