Khatri-Rao Products for Operator Matrices Acting on the Direct Sum of Hilbert Spaces

Abstract

We introduce the notion of Khatri-Rao product for operator matrices acting on the direct sum of Hilbert spaces. This notion generalizes the tensor product and Hadamard product of operators and the Khatri-Rao product of matrices. We investigate algebraic properties, positivity, and monotonicity of the Khatri-Rao product. Moreover, there is a unital positive linear map taking Tracy-Singh products to Khatri-Rao products via an isometry.

1. Introduction

In matrix theory, there are various matrix products which are of interest in both theory and applications, such as the Kronecker product, Hadamard product, and Khatri-Rao product; see, for example, [1–3]. Denote by the set of -by- complex matrices and abbreviate to . Recall that the Kronecker product of and is given by The Hadamard product of is defined by the entrywise product Now, let and be complex matrices partitioned into blocks and for each , (the sizes of and may be different). Then, the Khatri-Rao product [4] of and is defined by When and are nonpartitioned (i.e., each has only one block), their Khatri-Rao product is just their Kronecker product. If and are entrywise partitioned (i.e., each block is a matrix), then their Khatri-Rao product is their Hadamard product. Interesting algebraic, order, and analytic properties of this product were studied in the literature; see, for example, [5–12]. Their applications in statistics, computer science, and related fields can be seen, for example, in [13, 14].

The tensor product of Hilbert space operators is a natural extension of the Kronecker product to the infinite-dimensional setting. Let , , , and be Hilbert spaces. Recall that the tensor product of two operators and is the unique bounded linear operator from into such that, for all and ,

In this paper, we generalize the tensor product of operators to the Khatri-Rao product of operator matrices acting on a direct sum of Hilbert spaces. We investigate fundamental properties of this operator product. Algebraically, this product is compatible with the addition, the scalar multiplication, the adjoint operation, and the direct sum of operators. By introducing suitable operator matrices, we can prove that there is a unital positive linear map taking the Tracy-Singh product to the Khatri-Rao product . Hence, the Khatri-Rao product can be viewed as a generalization of the Hadamard product of operators. Moreover, positivity, strict positivity, and operator orderings are preserved under the Khatri-Rao product. Our result extends well-known results for Khatri-Rao products of complex matrices (see [4, 9, 15, 16]).

This paper is organized as follows. In Section 2, we provide some preliminaries about Tracy-Singh products for operators. These facts will be used in Sections 4 and 5. In Section 3, we introduce the Khatri-Rao product for operator matrices and deduce its algebraic properties. Section 4 explains how the Khatri-Rao product can be regarded as a generalization of the Hadamard product. Section 5 discusses positivity and monotonicity of Khatri-Rao products.

2. Preliminaries on Tracy-Singh Products for Operators

Throughout, let , , , and be complex separable Hilbert spaces. When and are Hilbert spaces, denote by the Banach space of bounded linear operators from into , and abbreviate to . If an operator satisfies , we write . For self-adjoint operators , we write to mean that is a positive operator, while means that .

Decompose where all , , , and are Hilbert spaces. For each , , let and be the canonical embeddings. For each and , let and be the orthogonal projections. Two operators and can thus be represented uniquely as operator matrices where and for each , , , and . We define the Tracy-Singh product of and to be the bounded linear operator from to expressed in a block-matrix form Basic properties of the Tracy-Singh product are listed below.

Lemma 1. The Tracy-Singh product is a bilinear map for operators. It is positive in the sense that if and , then .

3. Compatibility of Khatri-Rao Products with Algebraic Operations

In this section, we define the Khatri-Rao product for operator matrices and show that this product is compatible with certain algebraic operations of operators.

From now on, fix the following orthogonal decompositions of Hilbert spaces: That is, we fix how to partition any operator matrix in and . We now extend the Khatri-Rao product of matrices [4] to that of operators on a Hilbert space.

Definition 2. Let and be operators partitioned into matrices according to decomposition (8). We define the Khatri-Rao product of and to be the bounded linear operator from to represented by the block-matrix

If both and are block operator matrices, then is . When and for all , , the Khatri-Rao product is the Hadamard product of complex matrices.

Next, we shall show that the Khatri-Rao product of two linear maps induced by matrices is just the linear map induced by the Khatri-Rao product of these matrices. Recall that, for each and , the induced maps, are bounded linear operators. We identify with together with the canonical bilinear map for each .

Lemma 3. For any and , one has

Proof. Recall that the Kronecker product of matrices has the following property (see, e.g., [3]): provided that all matrix products are well defined. It follows that, for any and , The uniqueness of tensor products implies that .

Proposition 4. For any complex matrices and partitioned in block-matrix form, one has

Proof. Recall that the th block of the matrix representation of is . By Lemma 3, we obtain .

The next result states that the Khatri-Rao product is bilinear and compatible with the adjoint operation.

Proposition 5. Let and be operator matrices, and let . Then,

Proof. Since and , we obtain The fact that for all , together with the left distributivity of the tensor product over the addition implies Similarly, we obtain property (17). Since for all , , we get Similarly, .

By property (15), the self-adjointness of operators is closed under taking Khatri-Rao products; that is, if and are self-adjoint, then so is . The next proposition shows that, in order to compute the Khatri-Rao product of operator matrices, we can freely merge the partition of each operator.

Proposition 6. Let and be operator matrices represented according to decomposition (8). We merge the partition of to be , where , are given natural numbers such that and . Here, each operator is of block in which the th block of is the th block of , where Similarly, we repartition , where each operator is of block in which the th block of is the th block of . Then, That is, each th block of is just .

Proof. Write , where is block operator matrix such that the th block of is the th block of . We know that the th block of is . Then, Similarly, we have for all and .

Recall that the direct sum of , is defined to be the operator matrix The next result shows that the Khatri-Rao product is compatible with the direct sum of operators.

Proposition 7. For each , let and be compatible operator matrices. Then,

Proof. It follows directly from Proposition 6.

In summary, the Khatri-Rao product is compatible with fundamental algebraic operations for operators.

4. The Khatri-Rao Product as a Generalization of the Hadamard Product

In this section, we explain how the Khatri-Rao product can be viewed as a generalization of the Hadamard product. To do this, we construct two isometries which identify which blocks of the Tracy-Singh product we need to get the Khatri-Rao product.

Fix a countable orthonormal basis for . Recall that the Hadamard product of and in is defined to be the operator in such that for all . More explicitly, it was shown in [15] that where is the isometry defined by for all . When and is the standard ordered basis of , the Hadamard product of two matrices reduces to the entrywise product (2).

We now extend selection matrices in [9] to selection operators. Fix an ordered -tuple of Hilbert spaces endowed with decomposition (8). For each , consider the operator matrix where is the identity operator if and the others are zero operators. Similarly, for , we define the operator matrix where is the identity operator if and the others are zero operators. Now, construct We call and selection operators associated with the ordered tuple . Notice that depends only on the ordered tuple and how we decomposed and . The operator depends on and how we decomposed and . For instance, an ordered tuple with decompositions has the following selection operators: In the case of and , construction (31) gives If is the ordered pair of selection operators associated with the ordered tuple with decompositions given by (8), then is the ordered pair of selection operators associated with the ordered collection with the same decompositions.

Lemma 8. Let and be selection operators defined by (31). Then, for , (i); that is, is an isometry;(ii).

Proof. A direct computation shows that and . We know that is an block operator matrix which consists only of zero and identity operators. More precisely, the th block of is the identity operator and for . Then, Since for all , we have . Similarly, .

Next, we relate the Khatri-Rao and the Tracy-Singh product of operators.

Theorem 9. For any operator matrices and , one has where and are the selection operators defined by (31). If and , and , then where is the selection operator defined by (34).

Proof. Let denote the th column of for . Then, we have If and , then and (36) becomes (37).

We mention that Theorem 9 is an extension of both [9, Theorem ] and result (28) in [15].

Remark 10. If we partition and into row operator matrices, we have If both and are column operator matrices, then

Comparing (28) and (37), Theorem 9 shows that the Khatri-Rao product can be regarded as a generalization of the Hadamard product.

Recall that a map between two -algebras is said to be positive if preserves positive elements. The map is unital if preserves the multiplicative identity.

Corollary 11. There is a unital positive linear map such that for any and .

Proof. Define , where is the selection operator defined by (37) in Theorem 9. The map is clearly linear and positive. The map is unital since is an isometry (Lemma 8).

Corollary 11 provides a natural way to derive operator inequalities concerning Khatri-Rao products from existing inequalities for Tracy-Singh products.

The next result extends [16, Corollary ] to the case of Khatri-Rao and Tracy-Singh products of operators.

Corollary 12. Let and be operators in and , respectively. Then,

Proof. Using the fact that and if , where , we compute By applying Theorem 9, we get Similarly,

5. Positivity and Monotonicity of Khatri-Rao Products

In this section, we show that the Khatri-Rao product preserves positivity and strict positivity. It follows that operator orderings are preserved under Khatri-Rao products.

Theorem 13. Let and be operator matrices. If and , then .

Proof. It follows from the positivity of the Tracy-Singh product (Lemma 1) and Theorem 9.

The next result provides the monotonicity of Khatri-Rao product which is an extension of [9, Theorem ] to the case of operators.

Corollary 14. Let and . If and , then .

Proof. Applying Proposition 5 and Theorem 13 yields Thus, .

Now, we will develop the result of [9, Theorem ] to the case of Khatri-Rao product of operators.

Theorem 15. Let and be operator matrices. If and , then .

Proof. The strict positivity of and the spectral theorem imply the existence of an increasing sequence of closed subspaces of such that, for each , for each . Let be the orthogonal projection onto for each . There are similar subspaces and orthogonal projections for the operator . Then, for each , we have and and hence by Corollary 14. Since the union of the subspaces in and of the subspaces in is dense, it follows that, for any , there is for which . Hence, This shows that .

Corollary 16. Let and . If and , then .

Proof. The proof is similar to that of Corollary 14. Instead of Theorem 13, we apply Theorem 15.

Finally, we mention that, by using the results in this paper, we can develop further operator identities/inequalities parallel to matrix results for Khatri-Rao products.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was supported by the Thailand Research Fund. The second author would like to thank the Thailand Research Fund for the financial support.