The central tendency for a set of matrices, where each matrix is positive definite, presents a unique challenge. Unlike simple scalar averages, the averaging process must ensure that the resulting matrix also retains the positive definite property. Several methods exist, each with distinct characteristics. A simple arithmetic mean may not always result in a positive definite matrix. Therefore, alternatives such as the Riemannian mean or geometric mean are often preferred. For example, consider two positive definite matrices, A and B. The arithmetic mean is (A+B)/2, while the geometric mean involves matrix exponentiation and logarithms, ensuring the result’s positive definiteness.
The computation of a central representative within a set of positive definite matrices holds significance in various fields. In diffusion tensor imaging, these matrices represent the diffusion properties of water molecules in biological tissues. Averaging these matrices allows for the reduction of noise and the extraction of representative diffusion characteristics within a region of interest. Historically, the development of appropriate averaging techniques has been driven by applications in signal processing, machine learning, and control theory, where positive definite matrices arise in covariance estimation, kernel methods, and system stability analysis. The use of appropriate mean computation ensures robustness and accuracy in these applications.
