Selçuk J. Appl. Math. Selçuk Journal of Vol. 8. No.1. pp. 3-8 , 2007 Applied Mathematics
Matrix Trace Results on the Hadamard and Khatri-Rao Products Mustafa Özel1 and Turgut Özi¸s2
1Division of Mathematics, Dokuz Eylül University, Engineering Faculty, Department
of Geophysics Tınaztepe, Buca, ˙Izmir 35160, Turkey; e-mail:mustafa.ozel@ deu.edu.tr
2Department of Mathematics, Faculty of Science, Ege University, Bornova, ˙Izmir
35100, Turkey;
e-mail:turgut.ozis@ ege.edu.tr
Received : August 13, 2006
Summary.Some matrix trace consequences involving the Hadamard and Khatri-Rao products of the matrices A and B are established by using the vec-operator. A similar conclusion between the Hadamard and Khatri-Rao products and a simple proof of the conclusion as in [7] are presented
Key words:Hadamard product; Khatri-Rao product; vec-operator. 2000 Mathematical Subject Classification 15A24, 15A69
1. Introduction
Hadamard, Khatri-Rao and Kronecker products have been studied and applied widely in matrix theory [3,4,5,7]. Also recently for partitioned matrices, some inequalities involving Khatri-Rao and Tracy-Singh products were given in [4,7]. Section 2 is devoted to some definitions of the above-mentioned products and elementary results. In section 3, a further study of the trace of partitioned matrices was carried out using the results of the Hadamard and Khatri-Rao products. A conclusion between the Hadamard and Khatri-Rao products of two matrices was established in Theorem 7. Finally, a simple proof of it was presented.
2. Basic Definitions and Theorems
In this section, we shall state some basic definitions and theorems, which will be needed in the sequel. For detail and proofs we refer to Refs. [1,2,3,4,5,6].
Definition 1Let = [] be an × matrix and = [] be a × matrix.
The Kronecker product of A and B is an × matrix which is defined to be the partitioned matrix
⊗ = []
where = 1 2 and = 1 2
Definition 2 Let = [] and = [] be each an × matrix; then the
Hadamard product of A and B is an × matrix of elementwise products ∗ = []
where = 1 2 and = 1 2
Definition 3 Let = [] be partitioned with of order × matrix
and = [] be partitioned with of order × matrix; then the
Khatri-Rao product of A and B, which we write as A¯B, is as a matrix of order (P) × (P) defined by
¯ = [⊗ ]
where ⊗ is of order ×
Hadamard and Kronecker products have the relation as
(1) ∗ = ( ⊗ )
for 2× selection matrix such that = . Note that is the ×
2matrix
(2) = [11 22 ]
where is the × matrix of zeros except for solely one in the ( ) th
position.
Definition 4The vec operator of A, vecA , is defined as
(3) = [1 2 ]
and is × 1vector formed from the columns of A, where is the columns of
A of order ×
(4) ( ⊗ ) = where tr(A) is the trace of A.
Theorem 2Let A and B be each × matrices; then
(5) () = ()
where is the transpose of A.
Theorem 3Let , , , , and be matrices as in the following; ( × ) ( × ) ( × ) ( × ) ( × ) then (6) () = ( ⊗ ) (7) () = ( ⊗ ) and (8) ( ) = ( ¯ )
where = [11 22 ] is an × 1vector and is an × identity
matrix.
Theorem 4For the matrices A, B, F, and G,
(9) ( ⊗ )( ¯ ) = ¯
3. Main Results
It is hereby some trace equalities of Kronecker, Hadamard, and Khatri-Rao products of matrices were established.
Lemma 1Let J be an 2× selection matrix; then
(10) = ¯
Proof.For = [11 22 ], we have = X =1 ⊗ = [11⊗ 1 22⊗ 2 ⊗ ]
where is an × 1 column vector which is “1” in the th and zero elsewhere is
called the unit vector. Then we write
= [11 22 ] ¯ [1 2 ]
and (10) is obtained.
Theorem 5Let A and B be any × matrices; then
(11) ( ∗ ) = [( ¯ )]( ⊗ )
Proof.Using the trace of (1), the commutative law of the trace of the product of two matrices, and (5)
(12) ( ∗ ) = ¡( ⊗ )¢=£¡¢¤ ( ⊗ ) is obtained.
Corollary 1Let A and B be × matrices; then
(13) ( ∗ ) = () [( ⊗ ) ] and
(14) ( ∗ ) = ()[ ¯ ( ⊗ )](2) where 2 is the 22identity matrix.
Proof.We have, using (12), (6) and (7) ( ∗ ) =£¡2¢¤
( ⊗ )
The second equality is proved in precisely the same way starting from (13) and using (8).
Now we will give a new theorem that shows the relation between Hadamard and Khatri-Rao products of two matrices.
(15) ( ∗ ) = ( ¯ )
Proof.To prove this theorem, first we write
( ∗ ) = [( ¯ )( ⊗ )]
using (5) in the right-hand side of (11). Then using (9) and (15) is obtained. Conversely, from Lemma 1 and (9) we write
( ¯ ) = [( ¯ )( ⊗ )] = "Ã X =1 ⊗ ! ( ⊗ ) #
Then, using distributive law
( ¯ ) = X
()
()
is obtained and the proof is completed.
Theorem 7There exist a real matrix of order 2× such that
=
and
(16) ∗ = ( ¯ )
for = [] and = [] of the same order × , be partitioned with
and of order × as all block submatrix, respectively and
(17) = { } where is the selection matrix of order 2×
Proof. Let be as in (17), using the equation = and the proof of
Theorem1 in [7] then ( ¯ ) =h(⊗ ) i =h¡(⊗ ) ¢ i = ∗ is obtained.
References
1. Brewer, J.W.( 1978): Kronecker products and matrix calculus in system theory, IEEE Transactions on Circuits and Systems, 25, no.9, 772-781.
2. Graybill, F.A.(1969): Introduction to Matrices with Applications in Statistics, (Belmont, CA: Wadsworth).
3. Koning, R.H., Neudecker, H. and Wansbeek, T.(1991): Block Kronecker products and the vecb operator, Linear Algebra Appl, 149, 165-184.
4.Liu, S.(1999): Matrix results on the Khatri-Rao and Tracy-Singh products, Linear Algebra Appl., 289, 267-277.
5. Mond, B. and Peæariæ, J.E.(1998): Inequalities for the Hadamard product of matrices, SIAM J. Matrix Anal. Appl., 19, no.1, 66-70.
6.Rao, C.R. and Mitra, S.K.(1971): Generalized Inverse of Matrices and Its Applica-tions, (New York: Wiley).
7.Zhang, X., Yang, Z. and Cao, C.(2002): Inequalities involving Khatri-Rao products of positive semi-definite matrices, Applied Mathematics E-Notes, 2, 117-124.
