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Results tagged with matrices
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user 117027
Questions where the notion matrix has an important or crucial role (for the latter, note the tag matrix-theory for potential use). Matrices appear in various parts of mathematics, and this tag is typically combined with other tags to make the general subject clear, such as an appropriate top-level tag ra.rings-and-algebras, co.combinatorics, etc. and other tags that might be applicable. There are also several more specialized tags concerning matrices.
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Calculate the Riemannian Hessian of Karcher mean problem on positive definite matrices
Consider a collection of positive definite matrices $\{A_1,...,A_n\}\in\mathbb{S}_{++}^d$, the Karcher mean of these matrices is given by (see (5.4) in [1]):
$$
\min_{X\in\mathbb{S}_{++}^d} f(X):=\frac …
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Calculate the Riemannian Hessian of Karcher mean problem on positive definite matrices
First denote $\mathrm{dist}(A, B):=\|\log(A^{-1/2}B A^{-1/2})\|_{F}$, which is the geodesic distance of two positive definite matrices (see (1), Chpater 6), then we just need to calculate the Euclidean … Positive definite matrices. Princeton university press, 2009.
(2) Boumal, Nicolas. An introduction to optimization on smooth manifolds. Cambridge University Press, 2023. …
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Difference between parallel transport and ambient projection
Consider a $d$-dimensional complete embedded Riemannian submanifold $(M,g)$ of a Euclidean space $\mathbb{R}^D$ (The major examples we consider are sphere and Stiefel manifold). Assume the sectional c …
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