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

# The inner product on symmetric positive definite matrices

The distance on the symmetric positive definite matrices $\SPD{n}$ is given by

where $\Log$ denotes the matrix logarithm and $\lVert\cdot\rVert$ is the Frobenius norm of the matrix.

### Matlab Documentation

 1 2 3 4 5 6 7 8 9 10 11 % d = dist(x,y) compute the distance between x,y from P(n). % % INPUT % y,x : two points (matrices) or sets of points (matrices) % on P(m) % % OUTPUT % d : resulting distances of each pair of points of p,q. % --- % Manifold-Valued Image Restoration Toolbox 1.0 % R. Bergmann ~ 2014-10-19 | 2015-04-11