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

# The orthonormal basis in a tangent space

This function computes an ONB and correspoinding curvature coefficients, belonging to the parallel transportet orthonormal frame and diagonalizes the curvature tensor along $\geo{x}{y}$. The set of vectors in a tangent space $T_x\SPD{n}$ added as last dimension of the array and consists of $d=\frac{n(n+1)}{2}$ vectors.

The following derivation of the orthonormal frame follows . Let $\xi=\log_xy\in T_x\SPD{n}$. We denote its eigenvalues by $\lambda_1,\ldots,\lambda_n$ and an orthonormal basis of eigenvectors by $\vect{v}_1,\ldots,\vect{v}_n\in\mathbb R^n$. Then we can write

For ease of notation we introduce the indexing system $\mathcal I = \{(i,j) : i=1,\ldots,n; j = i,\ldots,n\}$

Then the matrices

form an orthonormal basis of $T_x {\mathcal P}(r)$.

The corresponding coefficients characterizing curvature (see Jacobi fields ) are

### Matlab Documentation

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% [Xi,k] = TpMONB(x,y) Compute an ONB in TpM and curvature
% coefficients corresponding to the transported frame along g(.,x,y)
%
% INPUT
%     p : base point( sets)
% OPTIONAL:
%     q : directional indicator( sets) for the first vector(s).
% OUTPUT
%    W : orthonormal bases ( n x n x SetDim x Dimension )
%    k : (optional) curvature coefficients (Dimension x SetDim)
% ---
% MVIRT R. Bergmann, 2017-12-03