Scaling-Rotation Distance and Interpolation of Symmetric Positive-Definite Matrices

We introduce a new geometric framework for the set of symmetric positive-definite (SPD) matrices, aimed at characterizing deformations of SPD matrices by individual scaling of eigenvalues and rotation of eigenvectors of the SPD matrices. To characterize the deformation, the eigenvalue-eigenvector decomposition is used to find alternative representations of SPD matrices and to form a Riemannian manifold so that scaling and rotations of SPD matrices are captured by geodesics on this manifold. The problems of nonunique eigen-decompositions and eigenvalue multiplicities are addressed by finding minimal-length geodesics, which gives rise to a distance and an interpolation method for SPD matrices. Computational procedures for evaluating the minimal scaling-rotation deformations and distances are provided for the most useful cases of $2 \times 2$ and $3 \times 3$ SPD matrices. In the new geometric framework, minimal scaling-rotation curves interpolate eigenvalues at constant logarithmic rate, and eigenvectors at...

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