A Multiplicative Operation on Matrices with Entries in an Arbitrary Abelian Group Cyrus Hettle (cyrus.h@uky.edu) Robert P. Schneider (robert.schneider@uky.edu) University of Kentucky Abstract We define a new operation on ordered pairs of matrices called formal multiplication that depends on only one operation between matrix entries, instead of the usual two operations in a ring. This new operation allows us to generalize the theory of matrices to matrices with entries in an arbitrary abelian group, instead of a ring. Some results from traditional matrix algebra still hold in this theory, as well as a number of new identities. We consider matrices with entries in an arbitrary abelian group under addition, and we let denote this set of matrices. We note from the outset that any commutative group operation may be substituted for addition in the following study with similar results. Suppose we wish to multiply matrices in. In the usual matrix multiplication, if for matrices, then allowing to denote the entry of C in the th row and th column we have However, we have only a single operation in the group so we cannot carry out both multiplication and addition between the entries of and ; therefore matrix multiplication is not well-defined. We define a new operation on matrices in that we call formal multiplication represented by the symbol, which resembles matrix multiplication but uses only addition on the entries of matrices in. We define the formal product,, to be the matrix having entries For example, considering the formal product of two matrices with entries in, or we have
2 We observe that instead of considering individual entries of the matrices, because G is abelian, we can instead consider row and column sums, to simplify our calculations. Thus, let us define that for an matrix the th row and th column sums will be respectively denoted and where, given by the sums Then we can rewrite the entries of as Therefore, for formal multiplication of matrices, we need only consider objects (the row sums of and column sums of ) as opposed to objects (individual entries of both and ) in our computations of the entries. In addition, we define the operation we call formal addition of matrices, which in the present study is identical to the usual matrix addition; in the more general case where the group operation is not addition, the same definition applies with appropriate changes in notation. We define the formal sum,, to be the matrix having entries For example, considering the formal sum of two matrices with entries in, i.e. two matrices in we have the expected result We note that we have associativity in the group, so we may write and in general when addition is carried out times. Then as one might expect, if we take the formal sum of a matrix with itself we denote this operation by such that Similarly, such that, and in general we write when the operation is carried out times on a single matrix such that This
3 standard product notation is not to be confused with a formal product matrices. between two As is abelian, we also have from the above considerations that Note that with the formal addition operation between matrices, we are allowing an extension to the set of the operation initially defined on the group. The same extension will apply in the case of a group operation other than addition. Having defined the formal multiplication and addition operations, let us now consider a few properties of the resulting matrices. PROPOSITION 1. The set of matrices with entries in is closed under formal multiplication. Proof. Let Then is an matrix. Because is a group, all, and hence all. Therefore is an matrix with entries in, and. We define any set that consists only of matrices to be trivial; such a set PROPOSITION 2. has unique properties. Proof. We show that satisfies the hypotheses defining a group. (i) is closed by Proposition 1. (ii) Let. Then (iii) (iv) PROPOSITION 3.
4 Proof. We define isomorphism are satisfied by f: Then the following properties defining an (i) f is injective, for (ii) f is surjective, for given (iii) f is a homomorphism, for given, Therefore f is an isomorphism and the trivial set The following propositions apply only to non-trivial sets. PROPOSITION 4. Formal multiplication over any non-trivial set. is not commutative Proof. Let 0 be the identity in and be some non-identity element in. Then we observe that Therefore we have a counterexample and Matrices in are not associative under formal multiplication, unlike matrices under the usual matrix multiplication. PROPOSITION 5. Formal multiplication over any non-trivial is not associative.
5 Proof. Again we provide a counterexample that can be easily extended. As before, let 0 be the identity in G and let a be some non-identity element in G. Then This example can be extended to any non-trivial. PROPOSITION 6. There is no formal-multiplicative identity in any non-trivial set. Proof. As before, let 0 be the identity in G and let a be some non-identity element in G. Consider the matrix in, and suppose there exists some matrix such that. Then Thus But then Therefore there exists no such matrix, and hence no identity in THEOREM 7. No non-trivial set is a group. Proof. By Propositions 5 and 6, a non-trivial therefore is not a group. is not associative and has no identity; Recall that if is any matrix,. We define the transpose of a matrix in the same way. THEOREM 8. In all Proof. We have that
6 Then COROLLARY 9. In all then we have the identity Proof. COROLLARY 10. In all Proof. Let element in ), that is, denote the zero matrix having all its entries being zero (or the identity The next two propositions depend on the observations that and, similarly, that PROPOSITION 11. Let ; then we have the identity Proof. Examining each entry of, we see that The following proposition resembles distributivity in the usual matrix algebra.
7 PROPOSITION 12. Let ; then we have the identities and Proof. We prove the first identity by proceeding entry-wise. We have The second identity follows from a similar proof. We now define a useful tool for addressing certain problems of formal multiplication that we call the total sum of A, notated, to be the sum of all the entries of, i.e. The total sum itself has a number of nice properties detailed below, where we always take. PROPOSITION 13. We have the identity Proof. Using the above definition of the total sum and the properties of formal addition, we have
8 PROPOSITION 14. We have the identity Proof. We have THEOREM 15. The total sum is a linear transformation. Proof. Propositions 13 and 14 provide the two conditions for a linear transformation. PROPOSITION 16. We have the identity Proof. THEOREM 17. Proof.
9 Using the total sum, we have the following criterion for whether a matrix be written as the formal product of two matrices in, that is, whether can be factored under formal multiplication. COROLLARY 18. For, if there is no, then cannot be written as for any. Proof. Suppose that is the product of two matrices in ; then it follows from Theorem 17 that with. This is equivalent to the statement of the present corollary, which is the contrapositive of the preceding sentence. can We have found a number of other criteria for discerning whether any rectangular matrix having entries in an arbitrary group can be factored as the formal product of two other matrices under formal multiplication, including an if-and-only-if theorem by which we can construct matrices such that. These criteria will be detailed in a more thorough report. The following propositions connect the factorization of matrices to their determinants, as calculated in the usual manner, when. LEMMA 19. If can be written as the product of two other matrices in, then any minor of created by removing the th row and th column can also be written as the product of two other matrices in. Proof. Let such that. Then consider the matrices where is defined by deleting the th row of, adding each entry to the entry directly to the left and deleting the th column. This yields an matrix with the same row sums as, except that row has been deleted. Likewise we define by removing the th column of, adding to the entry directly above and deleting the th row. This yields an matrix with the same column sums as, except that column has been removed. Then the minor of created by removing the th row and th column is the product of and, and hence can be written as the product of two other matrices in. THEOREM 20. Let for. If can be written as the product of two other matrices in, then has no inverse in the set of matrices with elements in equipped with regular matrix multiplication. Proof. We proceed by induction. Suppose that can be written as the product of two other matrices in Clearly there is such a matrix, for instance
10 Then there exist such that Let respectively, and respectively. Then We compute Therefore C has no inverse in the set of matrices with elements in matrix multiplication. equipped with regular Suppose now that, and that the theorem holds for. Then can be written as the product of two other matrices in is a linear combination of the determinants of submatrices of C. However, by Lemma 19, each of these minors of C is reducible and by the induction hypothesis has determinant 0. Therefore and C has no inverse in the set of matrices with elements in equipped with regular matrix multiplication. COROLLARY 21. Let for. If then cannot be written as for any. Proof. As with the proof of Corollary 18, we note from Theorem 20 that if is the product of two matrices in then it follows that. This is equivalent to the statement of the present corollary, which is the contrapositive of the preceding sentence. In the following propositions, we focus our attention on sets where for instance, the set of all matrices with entries in. These matrices have fortunate cancellations that lead to additional special properties, of which we note a couple here.
11 We take formal exponentiation of a matrix A such that general where the formal products are carried out from left to right., and in PROPOSITION 22. For Proof. We have that Therefore Similarly Using Theorem 17 and the fact that, we find Then but Therefore, for all entries of and Proposition 22 follows.
12 We observe that in the case is even,, we can show in a similar manner to the second expression of the above proof therefore we have the following, simpler expression for Then PROPOSITION 23. For, if then we have the identity Proof. and Equating the left-most term in each of the above chains of identities completes the proof. The reader will note the similarity of Proposition 23 to Corollary 10. Other properties of matrices in when will be addressed in a future report. The propositions proved in this study are special cases of a general theory of formal matrix operations to be laid out in a more complete report, which we have worked out fully in our private notes. Of course, we might substitute other abelian groups in the foregoing proofs, with similar results. For example, take the group, the reals excluding zero under multiplication. In this case we write the formal sum as instead of, as we wish to use the group operation as the operation between the matrices. Then the entries of are of the form
13 where the row sum and column sum in this case are products instead of actual sums. Moreover, the considerations outlined above generalize beyond sets of square matrices, to include matrices where and are not necessarily equal. We denote such a set of rectangular matrices by ; in this notation. In this more general setting, formal multiplication and formal addition take similar forms as defined at the outset of this report. Take, with abelian; then we have with entries defined by where the appropriate group operation should be substituted for the summation notation. For formal addition we require matrices of the same sizes; if we take then has entries defined by, as in the case of square matrices. Most of the definitions, properties and propositions given in this report generalize to rectangular matrices with entries in abelian groups. For example, the total sum of a matrix takes the general form Then if, Theorem 17 generalizes to This identity suggests a link between the total sum of the product of rectangular matrices and well-known theorems involving the greatest common denominator in number theory. In fact we find, similarly to Corollary 18, that if there is no element, then cannot be factored into a formal product for any matrices. We note that the propositions concerning determinants do not generalize, as they only make sense for square matrices with numerical entries. If we strictly adhere to the convention with formal addition that always yields entries of the form but not necessarily of the form that is, the left-right order of the matrices in the sum is reproduced in the order of the sum of the entries then we may even allow the extension of the group operation (whether addition or another operation) to the set of matrices when the operation on is not commutative. The
14 same is true for formal multiplication: if the left-right order of the matrices in the product is reproduced in the computation of the entries, i.e. we define the entries of strictly by then the operation over needs not be commutative. We leave the proofs of these and other general considerations for a future, more in-depth report, noting that our investigations extend even to sets of matrices having entries from any set whatsoever (not only a group) with any binary operation at all defined between its elements; such sets have been discussed by A. Bridy and D. Dynerman. Even with such extremely general sets of matrices, some of the structure and theorems found above for example, the identity are retained so long as the operation is commutative.