Block Diagonalization of Quaternion Circulant Matrices with Applications to Quaternion Tensor Singular Value Decomposition
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It is well-known that a complex circulant matrix can be diagonalized by a discrete Fourier matrix with imaginary unit i. The main aim of this paper is to demonstrate that a quaternion circulant matrix cannot be diagonalized by a discrete quaternion Fourier matrix with three imaginary units i,j,k. Instead, a quaternion circulant matrix can be block-diagonalized into 1-by-1 block and 2-by-2 block matrices by permuted discrete quaternion Fourier transform matrix. With such a block-diagonalized form, the inverse of a quaternion circulant matrix can be determined efficiently, similar to the inverse of a complex circulant matrix. We make use of this block-diagonalized form to study quaternion tensor singular value decomposition of quaternion tensors where the entries are quaternion numbers. Numerical examples of color videos as third-order quaternion tensors are employed to validate the effectiveness of quaternion tensor singular decomposition in terms of compression and calculation capabilities.
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