  We introduce a new geometric framework for the set of symmetric
positive-definite (SPD) matrices, aimed to characterize 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 non-unique
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 to evaluate 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 constant angular rate. In the context of
diffusion tensor imaging, this results in better behavior of the trace,
determinant and fractional anisotropy of interpolated SPD matrices in typical
cases.
