550 research outputs found
Geometric Distance Between Positive Definite Matrices of Different Dimensions
We show how the Riemannian distance on n++, the cone of n×n real symmetric or complex Hermitian positive definite matrices, may be used to naturally define a distance between two such matrices of different dimensions. Given that n++ also parameterizes n-dimensional ellipsoids, and inner products on ℝn, n×n covariance matrices of nondegenerate probability distributions, this gives us a natural way to define a geometric distance between a pair of such objects of different dimensions.ER
Inverting a complex matrix
We analyze a complex matrix inversion algorithm proposed by Frobenius, which
we call the Frobenius inversion. We show that the Frobenius inversion uses the
least number of real matrix multiplications and inversions among all complex
matrix inversion algorithms. We also analyze numerical properties of the
Frobenius inversion. We prove that the Frobenius inversion runs faster than the
widely used method based on LU decomposition if and only if the ratio of the
running time of the real matrix inversion to that of the real matrix
multiplication is greater than . We corroborate this theoretical result by
numerical experiments. Moreover, we apply the Frobenius inversion to matrix
sign function, Sylvester equation, and polar decomposition. In each of these
examples, the Frobenius inversion is more efficient than inversion via
LU-decomposition
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