$\DeclareMathOperator{\arccosh}{arccosh} \DeclareMathOperator*{\argmin}{arg\,min} \DeclareMathOperator{\Exp}{Exp} \newcommand{\geo}[2]{\gamma_{\overset{\frown}{#1,#2}}} \newcommand{\geoS}{\gamma} \newcommand{\geoD}[2]{\gamma_} \newcommand{\geoL}[2]{\gamma(#2; #1)} \newcommand{\gradM}{\nabla_{\M}} \newcommand{\gradMComp}[1]{\nabla_{\M,#1}} \newcommand{\Grid}{\mathcal G} \DeclareMathOperator{\Log}{Log} \newcommand{\M}{\mathcal M} \newcommand{\N}{\mathcal N} \newcommand{\mat}[1]{\mathbf{#1}} \DeclareMathOperator{\prox}{prox} \newcommand{\PT}[3]{\mathrm{PT}_{#1\to#2}#3} \newcommand{\R}{\mathbb R} \newcommand{\SPD}[1]{\mathcal{P}(#1)} \DeclareMathOperator{\Tr}{Tr} \newcommand{\tT}{\mathrm{T}} \newcommand{\vect}[1]{\mathbf{#1}}$

# The inner product on symmetric positive definite matrices

The affine invariant metric on the symmetric positive definite matrices is defined for $x\in\SPD{n}$ and $\xi,\nu\in T_x\SPD{n}$ as

where $\Tr$ is the trace of a matrix.

### Matlab Documentation

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% dot(x,xi,nu) inner product of two tangent vectors in TxP(m)
%
% INPUT
%     x  : a point(Set) in P(n)
%     xi : a first tangent vector( set) to (each) x
%     xu : a secod tangent vector( set) to (each) x
%
% OUTPUT
%     ds : the corresponding value(s) of the inner product of (each triple) V,W at X
%
% ---
% Manifold-Valued Image Restoration Toolbox 1.0
% R. Bergmann ~ 2015-01-29 | 2015-04-11