Geometric means of structured matrices

The geometric mean of positive definite matrices is usually identified with the Karcher mean, which possesses all properties—generalized from the scalar case—a geometric mean is expected to satisfy. Unfortunately, the Karcher mean is typically not structure preserving, and destroys, e.g., Toeplitz and band structures, which emerge in many applications. For this reason, the Karcher mean is not always recommended for modeling averages of structured matrices. In this article a new definition of a geometric mean for structured matrices is introduced, its properties are outlined, algorithms for its computation, and numerical experiments are provided. In the Toeplitz case an existing mean based on the Kähler metric is analyzed for comparison.