Let complex matrices $A$ and $B$ have the same sizes. Using the singular value decomposition, we characterize the $g$-inverse $B^{(1)}$ of $B$ such that the distance between a given $g$-inverse of $A$ and the set of all $g$-inverses of the matrix $B$ reaches minimum under the unitarily invariant norm. With this result, we derive additive and multiplicative perturbation bounds of the nearest per...

Low rank matrix approximations have many applications in different domains. In system theory it has been used in model reduction schemes, in system identification with outputerror models and in static errors-in-variables problems, for instance. The approximations are mostly performed using the singular value decomposition. This is optimal for all unitarily invariant matrix norms, such as the Fr...

Let Gm,n be the Grassmann space of m-dimensional subspaces of F. Denote by θ1(X ,Y), . . . , θm(X ,Y) the canonical angles between subspaces X ,Y ∈ Gm,n. It is shown that Φ(θ1(X ,Y), . . . , θm(X ,Y)) defines a unitarily invariant metric on Gm,n for every symmetric gauge function Φ. This provides a wide class of new metrics on Gm,n. Some related results on perturbation and approximation of subs...

In recent years, structured matrix recovery problems have gained considerable attention for its real world applications, such as recommender systems and computer vision. Much of the existing work has focused on matrices with low-rank structure, and limited progress has been made on matrices with other types of structure. In this paper we present non-asymptotic analysis for estimation of general...

Let m,n, p be positive integers such that m ≥ n + p. Suppose (A,B) ∈ Cm×n ×Cm×p, and let P(A,B) = {(E,F ) ∈ Cm×n ×Cm×p : there is X ∈ Cn×p such that (A− E)X = B − F}. The total least square problem concerns the determination of the existence of (E,F ) in P(A,B) having the smallest Frobenius norm. In this paper, we characterize elements of the set P(A,B) and derive a formula for ρ(A,B) = inf {‖[...

In this article we study the Heinz and Hermite-Hadamard inequalities. We derive whole series of refinements these inequalities involving unitarily invariant norms, which improve some recent results, known from literature. We also prove that if $A , B, X\in M_n(\mathbb{C})$ such $A$ $B$ are positive definite $f$ is an operator monotone function on $(0,\infty)$. Then \begin{equation*} |||f(A)X-...