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.
Highlights
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, [1,2,3]
The tensor product of Hilbert space operators is a natural extension of the Kronecker product to the infinitedimensional setting
For selfadjoint operators A, B ∈ B(H), we write A ⩾ B to mean that A−B is a positive operator, while A > B means that A−B > 0
Summary
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, [1,2,3]. Interesting algebraic, order, and analytic properties of this product were studied in the literature; see, for example, [5,6,7,8,9,10,11,12]. The tensor product of Hilbert space operators is a natural extension of the Kronecker product to the infinitedimensional setting. (A ⊗ B) (x ⊗ y) = Ax ⊗ By. 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. By introducing suitable operator matrices, we can prove that there is a unital positive linear map taking the Tracy-Singh product A ⊠ B to the Khatri-Rao product A ⊡ B.
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