
# The exponential map on symmetric positive definite matrices

The exponential map on the sphere $\SPD{n}$ is given for $x\in\SPD{n}$, $\xi\in T_x\SPD{n}$ by the formula

where $\Exp$ denotes the matrix exponential.

### Matlab Documentation

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% y = exp(x,xi) exponential map at x from P(n) towards xi in TxP(n)
%
% INPUT
%   x : a point or set of points on the manifold P(n)
%   xi : a point or set of point in the tangential spaces TxM
%
% OPTIONAL
%   t : shorten vectors V by factor t
%   (given as one value or an array of same length as number of Vs)
%
% OUTPUT
%   Y : resulting point(s) on P(m)
% ---
% Manifold-Valued Image Restoration Toolbox 1.0, R. Bergmann ~ 2015-01-20 | 2015-04-10