Inverse operations on tensor products of matrices
Date
2022
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Abstract
A variety of products of matrices arise by considering different algebraic structures – for example, linear transformations (matrix multiplication), product vector spaces which leads to entrywise products (known as the Hadamard or Schur product) and bilinear transformations (tensor products).
Inverse operations of linear transformations have been extensively studied in the literature, but inverse tensor products are less well known.
This dissertation considers these inverse operations from different perspectives, by focusing on characterising such operations and examining certain desirable properties. Primarily, by abiding to an algebraic perspective, quotients of vector spaces of (ms) × (nt) matrices are considered by characterising linear quotient functions. Requirements for such functions to satisfy desirable properties, in addition to linear properties, are considered. Additional quotients, which do not appear in the literature, are derived. Multiplicative (monoidal) quotients are also considered. These quotients only exist on restricted structures, and their limitations are briefly examined.
Lastly, by relaxing the requirement for a purely algebraic quotient and finitedimensional spaces, an analytic approach is considered by assessing a least squares minimisation of objects on reproducing kernel Hilbert spaces. In this method, Tikhonov regularisation is employed to ensure boundedness in obtaining inverse operations
Description
A dissertation submitted in fulfilment of the requirements for the degree of Master of Science to the Faculty of Science, University of the Witwatersrand, Johannesburg, 2022
Keywords
Inverse operations, Tensor products, Matrices