TRACES OF SCHUR AND KRONECKER PRODUCTS FOR BLOCK MATRICES

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1 Khayyam J. Math. DOI: /kjm TRACES OF SCHUR AND KRONECKER PRODUCTS FOR BLOCK MATRICES ISMAEL GARCÍA-BAYONA Communicated by A.M. Peralta Abstract. In this aer, we define two new Schur and Kronecker-tye roducts for block matrices. We resent some equalities and inequalities involving traces of matrices generated by these roducts and in articular we give conditions under which the trace oerator is sub-multilicative for them. Also, versions in the block matrix framework of results of Das, Vashisht, Taskara and Gumus will be obtained. 1. Introduction and reliminaries If we take two matrices A a k,j k,j and B b k,j k,j of the same size, with entries in the comlex or real field, their Hadamard roduct is just their elementwise roduct, that is, A B a k,j b k,j k,j. Since Schur rovided the initial studies about its roerties, it is widely known also as the Schur roduct. Horn, in 1990, gave a rofound insight on this roduct see [6]. In what follows, we will denote by M n,m X the sace of matrices of size n m with entries in X. If X is also a sace of matrices, we will use the exression block matrices to refer to the elements of M n,m X. Consider now A a k,j k,j M n,m C and B b k,j k,j M,q C. Their Kronecker roduct, Date: Received: 8 October 2018; Revised: 7 January 2019; Acceted: 16 January Mathematics Subject Classification. Primary 15A45; Secondary 47A50, 47L10, 15A16. Key words and hrases. Schur roduct, Kronecker roduct, trace, matrix multilication, inequalities. 1

2 2 I. GARCÍA-BAYONA denoted by A B, is defined as follows: a 1,1 B a 1,2 B a 1,m B a A B : 2,1 B a 2,2 B a 2,m B M n,mqc. a n,1 B a n,2 B a n,m B Both the Schur roduct and the Kronecker roduct are studied and alied in fields such as matrix theory, matricial analysis or statistics. For instance, the reader is referred to [7], where Magnus and Neudecker gave some results and statistical alications regarding the Schur and Kronecker roducts and to [8], where Persson and Poa used the Schur roduct as a tool in the area of matricial harmonic analysis to develo theories of matrix saces arallel to their scalar counterarts. The trace of a matrix A a k,j k,j M n n C is the sum of its diagonal elements, that is, tra a i,i. The reader is referred to the aers of Das and Vashisht see [3] and Taskara and Gumus see [9], where the authors investigated traces of Schur and Kronecker roducts. One of our goals in this aer will be to generalize some of those results in the context of block matrices, for certain versions of Schur and Kronecker roducts that we shall define now. Definition 1.1. Let A T k,j k,j M N M M n m C and B S k,j k,j M N M M n m C. We define the Schur roduct of A and B as A B : T k,j S k,j k,j, where T k,j S k,j denotes the classical Schur roduct of the matrices T k,j and S k,j. If m, n 1 or N, M 1, this roduct coincides with the classical Schur roduct. Although the revious definition is natural, other otions to define a Schur roduct for block matrices exist, of course. For examle, in [1, 2], we worked with a definition that involved the entry-wise comosition of oerators and roved some results with it in the field of matricial harmonic analysis for infinite matrices. In this aer, we will focus on the context of finite matrices. Consider now T M n,m C and B B k,j k,j M N,M M n,m C. We define a block Kronecker roduct of T and B as T B T B k,j k,j. Taking this into account, we can define our Kronecker roduct of two block matrices as follows. Definition 1.2. Let A T k,j k,j M N M M n m C and B S k,j k,j M P Q M n m C. We define their Kronecker roduct, A B, as A B : T k,j B k,j M NP MQ M n m C. Observe that this roduct is not commutative, but the Schur roduct for block matrices is. Also, note that if m, n 1, then this Kronecker roduct becomes the Kronecker roduct of matrices with comlex entries; and if P, Q, N, M 1, then

3 TRACES OF SCHUR AND KRONECKER PRODUCTS FOR BLOCK MATRICES 3 the classical Schur roduct is obtained. Again, we oint out that other natural definitions of a block Kronecker roduct exist see [5]. Some basic roerties satisfied by these roducts are the following ones. Let N, M, P, Q, n, m N. Proerty 1 The roducts and establish bilinear mas between saces of block matrices: : M N M M n m C M N M M n m C M N M M n m C and : M N M M n m C M P Q M n m C M NP MQ M n m C. Proerty 2 Associativity. A B C A B C. A B C A B C. Proerty 3 Distributivity with resect to the sum. A + B C A C + B C. A B + C A B + A C. A + B C A C + B C. A B + C A B + A C. Proerty 4 Mixed associativity. For every α C, αa B αa B A αb. αa B αa B A αb. Proerty 5 Commutativity of. A B B A. Since the sace of block matrices with the oerations + sum of matrices and roduct by scalar is a vector sace, it becomes an algebra when equied with due to roerties 2 4, and it becomes a commutative algebra when equied with instead, due to roerties On traces of block matrices In what follows, we will study equalities and inequalities involving traces of block matrices and the roducts defined above. From now on we will work in the context of square block matrices whose entries are also square matrices, with entries in R, and will abbreviate the notation for these matrices in the following way: M N M n : M N N M n n R. First of all, take into account that the trace of a block matrix is comuted by summing the traces of its diagonal elements. That is, if A T k,j k,j M N M n, then tra trt i,i T i,i l, l. Proosition 2.1. Let A M N M n and B M M M n with A T k,j k,j and B S k,j k,j. Then a If M N, then tra B T i,i l, ls i,i l, l.

4 4 I. GARCÍA-BAYONA b tra B T i,i l, ls j,j l, l. c If M N, then tra B tra B + Proof. a b tra B tra B tra B i,i c Follows from a and b., i j T i,i l, ls i,i l, l. trt i,i B T i,i l, ls j,j l, l. T i,i S i,i l, l trt i,i S j,j T i,i l, ls j,j l, l. Remark 2.2. Observe that although we know that given A and B square block matrices in general one has A B B A, a glance at art b in Proosition 2.1 shows that tra B is always equal to trb A. Of course, tra B trb A since the matrices coincide. Proosition 2.3. Let A, B M N M n. We have a tr A + B A B tra A trb B. b tr A ± B A ± B tra A ± 2 tra B + trb B. c tr A + B A B tra A trb B. d tr A ± B A ± B tra A ± 2 tra B + trb B. Proof. The four assertions are consequence of the roerties of mixed associativity and distributivity with resect to the sum that the Kronecker and the Schur roduct have, and also the linearity of the trace and Remark 2.2. Remark 2.4. Recall that the arithmetic mean of a sequence α l m is greater or equal than its geometric mean, that is, m α m 1 m l m α l. Proosition 2.5. Let N, and consider a finite sequence of block matrices A s s1 M N M n. Then, we have the following inequality: tr s1a s tr A s. s1

5 Proof. tr TRACES OF SCHUR AND KRONECKER PRODUCTS FOR BLOCK MATRICES 5 s1 A s A A tr A A tr 1 A i,i l, l + + A i,i s1 tr A s i,i l, l N s1a s i,i tr A A l, l A A s1a s i,i tr s1a s. i,i by Remark Trace sub-multilicativity and the saces M S N M n and M + N M n Now, we will exlore the relation between the value of the trace of Schur or Kronecker roducts of matrices and the roduct of the traces of the original matrices. First of all, recall that in the case of matrices with scalar entries, it is known that for the Kronecker roduct one has tra B tra trb. This is not the case for block matrices, neither for the roduct nor for the roduct, as the following examle shows. Examle 3.1. Consider the following matrices from M 2 M 2 : A , B Their block Schur roduct is the matrix A B M 2M 2,

6 6 I. GARCÍA-BAYONA and their block Kronecker roduct is the matrix A B M 4 M Then, tra 3, trb 4, but tra B 7 tra trb and tra B 6 tra trb. Examle 3.1 also revealed that the inequalities tra B tra trb and tra B tra trb are not true in general. However, we ask ourselves if the trace oerator might be submultilicative for these roducts under certain restrictions. The following two saces of block matrices will be relevant for that matter. Definition 3.2. Given N, n N, we define the following subsets of M N M n : M S NM n : {T k,j k,j M N M n / T k,k l, l 0, 1 l n}, M + N M n : {T k,j k,j M N M n / T k,k l, l 0, 1 k N, 1 l n}. Of course, M + N M n M S N M n. In the next theorem, we show that the trace oerator is actually submultilicative when acting on matrices that fulfill conditions related to the saces of matrices from Definition 3.2. Theorem 3.3. Let A T k,j k,j M N M n and B S k,j k,j M M M n. a If M N, A M S N M n, and B M + N M n, then k1 tra B tra trb. b If A M S N M n and B M S M M n, then tra B tra trb. Proof. a First, observe that since A M S N M n, then we have that T k,k l, l T k,k l, l tra. k1 k1 Also, since B M + N M n, we get the following estimation: su S i,i l, l i S i,i l, l S i,i l, l trb.

7 TRACES OF SCHUR AND KRONECKER PRODUCTS FOR BLOCK MATRICES 7 Therefore, tra B T i,i l, ls i,i l, l su S i,i l, l i tra trb. T i,i l, l by Proosition 2.1a b In a similar way, we get the estimation for the trace of the block Kronecker roduct: tra B T i,i l, ls j,j l, l N M T i,i l, l S j,j l, l su l S j,j l, l N T i,i l, l S j,j l, l tra tra trb. Corollary 3.4. a Let A 1, A 2,..., A m M + N M n. Then m tra 1 A 2 A m tra i. b Let B i M + N i M n, 1 i m. Then trb 1 B 2 B m Proof. a Note that for any 1 i m, one has m trb i. by Proosition 2.1b A 1 A 2 A i k,k l, l A 1 k,k l, l A 2 k,k l, l... A i k,k l, l 0 for all k, l such that 1 k N and 1 l n, since by hyothesis all matrices A i are in M +. Therefore, A 1 A 2 A i is in M + for each i. Now, the inequality follows from a combined use of this observation, art a of Theorem 3.3 and an induction argument. b Following the same line as in a, since the matrix B 1 B i is also in M + for each 1 i m because its diagonals are just Schur roducts of diagonals of matrices that are all of them in M + by hyothesis. This allows us to aly art b of Theorem 3.3, and an induction argument concludes the roof.

8 8 I. GARCÍA-BAYONA As a small alication of the revious inequality, we will take a look now at the trace of a version of the exonential of a block matrix based on our block Schur roduct. Let A M N M n be a block matrix. We define e A as follows: e A : j0 j A, where 0 A : I M N M n is the identity matrix for the block Schur roduct, with tri Nn. Observe that e A is well defined, since taking multilier norms, we have j A j A A j e A. j0 j0 Notice that in the second inequality we used the fact that the roduct endows the sace of multiliers from M N M n to M N M n with a structure of Banach algebra. Furthermore, it can be seen that the roduct also endows the sace of bounded linear oerators reresented by elements of M N M n with a structure of Banach algebra with the oerator norm see [4]. Now, by letting d Nn 1, the trace of e A can be bounded from above when A M + N M n as follows: tre A tr Nn + j0 j A j0 j0 tr j A 3.1 tra j d + e tra. by Corollary 3.4a Proosition 3.5. Let A, B M N M n such that A + B M + N M n, and let d Nn 1. Then tre A e B d + e tra e trb. Proof. Note that, since the roduct is commutative, we can write e A e B j0 m0 j0 l0 l0 1 l! j A j0 m A j B m! l m0 j B l l0 m0 l! m!l m! m A l m B l A + B l! e A+B. m A l m B m!l m! Using that and alying inequality 3.1 to A + B, we have tre A e B tre A+B

9 TRACES OF SCHUR AND KRONECKER PRODUCTS FOR BLOCK MATRICES 9 d + e tra+b d + e tra+trb by 3.1 d + e tra e trb. As a direct consequence of alying induction to Proosition 3.5, we obtain this corollary that gives an uer estimate for the trace of a finite Schur roduct of exonentials of matrices. Corollary 3.6. Let {A i } m M NM n such that m A i M + N M n, and let d Nn 1. Then, we have m tr m e A i d + e trai. 4. Trace inequalities combining both roducts Finally, we resent some results that give uer estimates for the traces of block matrices generated by combined Kronecker and Hadamard roducts, in terms of the trace of matrices where only one of the roducts is involved. The utility of these lies in the fact that it is easier to comute the latter ones. Theorem 4.1. Let A i, B i M + N M n, for 1 i m. Then a tr A 1 A 2 A m B 1 B 2 B m m tra i B i. b tr A 1 A 2 A m B 1 B 2 B m m tra i B i. Proof. First, observe that direct comutations show that the roduct and the roduct are linked by the following relation, that we shall use below: A B C D A C B D. 4.1 a We use an induction argument. For m 1, the result is obvious. Let us assume that the result is true for m s, and let us rove that it is also true for m s + 1. We can write tr A 1 A 2 A s+1 B 1 B 2 B s+1 tr A1 A 2 A s B 1 B 2 B s A s+1 B s+1. by 4.1 Now, by hyothesis, A s+1 B s+1 is in a sace M +. The matrix A 1 A 2 A s B 1 B 2 B s also belongs to the same sace, since all of its diagonal entries are Schur roducts of the diagonal entries of the matrices A i, B j which all of them were in M + by the hyothesis. Therefore, by using art b of Theorem 3.3 and induction hyothesis, we conclude tr A1 A 2 A s B 1 B 2 B s A s+1 B s+1

10 10 I. GARCÍA-BAYONA tr A 1 A 2 A s B 1 B 2 B s tra s+1 B s+1 s s+1 tra i B i tra s+1 B s+1 tra i B i. b By using art a of Theorem 3.3, the roof is analogous. by Theorem 3.3b Corollary 4.2. Let A i, B i M + N M n, for 1 i m. Then a tr A 1 A 2 A m B 1 B 2 B m tr m A i tr m B i. b tr A 1 A 2 A m B 1 B 2 B m tr m A i tr m B i. Proof. a tr A 1 A 2 A m B 1 B 2 B m tr A1 A 2 A m 1 A m B1 B 2 B m 1 B m tr A1 A 2 A m 1 B 1 B 2 B m 1 A m B m tra 1 A m 1 A m trb 1 B m 1 B m tr m A i tr m B i. by 4.1 by Theorem 4.1a b Follows by the same argument, but using art b of Theorem 4.1 instead. Acknowledgement. The author is thankful to the referees for their comments and also acknowledges the suort rovided by MINECO Sain under the roject MTM P and by MCIU Sain under the grant FPU14/ References 1. O. Blasco, I. García-Bayona, New saces of matrices with oerator entries, Quaest. Math. to aear. 2. O. Blasco, I. García-Bayona, Schur roduct with oerator-valued entries, Taiwanese J. Math., to aear. arxiv: [math.fa]. 3. P. K. Das, L. K. Vashisht, Traces of Hadamard and Kronecker roducts of matrices, Math. Al I. García-Bayona, Remarks on a Schur-tye roduct for matrices with oerator entries, submitted. 5. M. Günter, L. Klotz, Schur s theorem for a block Hadamard roduct, Linear Algebra Al R.A. Horn, The Hadamard roduct, Proc. Sym. Al. Math J.R. Magnus, H. Neudecker, Matrix Differential Calculus with Alications in Statistics and Econometrics, 2nd ed., Wiley, Chichester, UK, L.E. Persson and N. Poa, Matrix Saces and Schur Multiliers: Matriceal Harmonic Analysis, World Scientific, Hackensack, NJ, N. Taskara, I.H. Gumus, On the traces of Hadamard and Kronecker roducts of matrices, Selçuk J. Al. Math

11 TRACES OF SCHUR AND KRONECKER PRODUCTS FOR BLOCK MATRICES 11 Deartamento de Análisis Matemático, Universidad de Valencia, Burjassot, Valencia, Sain. address: