Finsler geometries on strictly accretive matrices

Abstract

In this work, we study the set of strictly accretive matrices, that is, the set of matrices with positive definite Hermitian part, and show that the set can be interpreted as a smooth manifold. Using the recently proposed symmetric polar decomposition for sectorial matrices, we show that this manifold is diffeomorphic to a direct product of the manifold of (Hermitian) positive definite matrices and the manifold of strictly accretive unitary matrices. Utilizing this decomposition, we introduce a family of Finsler metrics on the manifold and characterize their geodesics and geodesic distances. Finally, we apply the geodesic distance to a matrix approximation problem and also give some comments on the relation between the introduced geometry and the geometric mean of strictly accretive matrices as defined by Drury [1].

Keywords:

• Accretive matrices
• matrix manifolds
• Finsler geometry
• numerical range
• geometric mean

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Note that this is equivalent to the Toeplitz decomposition since if $A=\mathrm{\Re }\left(A\right)+i\mathrm{\Im }\left(A\right)$, then $\mathrm{\Re }\left(A\right)=H\left(A\right)$ and $\mathrm{\Im }\left(A\right)=\frac{1}{i}S\left(A\right)$, see [8, p. 7].

2 The naming used here is the same as in [9, p. 281], in contrast to [10].

3 In [24], these were called canonical angles.

4 To see this, note that a matrix A is normal if and only if H(A) and $\frac{1}{i}S\left(A\right)$ commute, see, e.g. [15, Thm. 9.1].

5 Note that the latter has unfortunately also been termed dissipative in the literature, see [9, p. 279].

6 In [7], the convention is that the Finsler structure is squared compared to the one in [34]. We follow the convention of the latter.

7 Note that functions Ψ fulfilling i)-v) are not necessarily symmetric gauge functions. As an example, consider Ψ(x, y) = (x² + 3xy + y²)² [7, Rem. 6]; this is not a symmetric gauge function since in general Ψ(x, y) ≠ Ψ(x, − y). Conversely, symmetric gauge functions do not in general fulfil (i)–(v). As an example, consider Φ(x, y) = max{|x|, |y|} [12, p. 138]; at any point $(x,y)\in {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}$ such that x>y, ∂_yΦ = 0 and hence condition (iv) is not fulfilled for this function.

8 To see this, take $V=P_r^{-1}{P}^2{P}_r^{-1}\in {\mathbb{P}}_n$ and $U=P_r^2$ in [12, Thm. 9.H.1.f], and use the fact that the eigenvalues of $UV=P_r{P}^2{P}_r^{-1}$ are invariant under the similarity transform $P_r^{-1}\cdot P_r$.

Additional information

Funding

This work was supported by the Knut and Alice Wallenberg foundation, Stockholm, Sweden, under grant KAW 2018.0349, and the Hong Kong Research Grants Council, Hong Kong, China, under project GRF 16200619.