ETD Collection

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    Inverse operations on tensor products of matrices
    (2022) van der Merwe, Francesca
    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
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    Gravitational description of the conformally invariant quantum mechanics of large matrices
    (2017) Hanmer, Jeffrey Thomas
    We study the collective field theory of a free multi-matrix model in the radial sector, which has an emergent 1/r2 term, and take the large N limit. We show that it is possible to generate 2−d metrics with generic dependence on the collective field Lagrange multiplier (μ) and potential and which are distinguished by the choice of the potential. The Lagrange multiplier is shown to depend on an induced scale parameter after an I.R. regularization and breaks scale invariance. The collective field sl(2, R) algebras of the free Hamiltonian and a related alternative compact operator only close in the absence of μ. We point out that the broken conformal symmetry is contained in the associated metrics which suggests that they are related to a Near-AdS2 geometry. We also comment on the resemblance of these metrics to black hole solutions.