To further reduce the number of recommended assembly tolerance types and to lay foundation for further assembly tolerance analysis and synthesis, a spatial relation layer is introduced into the polychromatic sets-based model and an assembly tolerance representation model based on spatial relations for generating assembly tolerance types is proposed. fact that the information of degrees of freedom cannot be processed in polychromatic sets.
Furthermore, the model cannot be directly applied to further assembly tolerance analysis and synthesis due to the. However, the number of recommended assembly tolerance types generated by the model is somewhat large for the same feature surfaces. The main advantage of polychromatic sets-based assembly tolerance representation model is that the number of feature types to be processed is larger. In particular, the inverse Artwish density plays a critical role in quantifying the performance of modern adaptive processors when utilizing sample covariance matrix estimates in snapshot-starved situations. As the underlying Artwish matrix is singular, the standard Moore-Penrose pseudo-inverse applies in the formulation of the inverse Artwish density. More importantly, the same methodology permits closed form evaluation of the real and complex forms of the Artwish and inverse Artwish densities that is, for situations involving rank deficiency in the quadratic result.
#UNITARY MATRIX FULL#
The standard real and complex forms of the Wishart and inverse Wishart densities are recovered for both marginal rank and full rank situations. The attendant transformations include conversions (in both directions) between matrices and their equivalent singular value decomposition (SVD) components, requiring knowledge of the Jacobian determinants associated with such conversions.
For all developments, novel direct derivations are employed, whereby the desired density is developed via a sequence of random variable transformations starting from the underlying Gaussian matrix density and ending with the desired quadratic result. This paper develops closed form results for the full range of Wishart and inverse Wishart matrix densities, which are the densities associated with quadratic and inverse quadratic transformations of zero mean Gaussian matrices. Extensions of such techniques in the presence of zero, repeated, or repeated zero singular values are also surveyed, but with less mathematical rigor. For the simplest case where all singular values are non-zero and not repeated, four different possible approaches are carefully considered, and each rigorously proven to generate unique (or almost unique) singular vector matrices.
#UNITARY MATRIX SERIES#
A series of alternative singular vector “normalization” techniques are proposed to alleviate the ambiguity and ensure that the two singular vector matrices are uniquely defined. In the complex case, the source of the ambiguity is the reference phase (that is, the definition of what constitutes zero phase) associated with each individual pair of left-hand and right-hand singular vectors in the real case, this ambiguity reduces to a choice of sign. However, the associated left-hand and right-hand singular vector matrices are not uniquely defined, even in the simplest case where no zero or repeated singular values exist. Once the singular values generated by the multi-dimensional transformation of singular value decomposition are arranged in decreasing (or other specified) order, the resulting singular value matrix is uniquely defined.
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