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Volume 10, Number 3, 2020, Pages 1107-1117                                                                DOI:10.11948/20190206
Unitarily invariant norm and $Q$-norm estimations for the Moore--Penrose inverse of multiplicative perturbations of matrices
Juan Luo
Keywords:Moore-Penorse inverse, multiplicative perturbation, unitarily invariant norm, $Q$-norm, norm upper bound.
      Let $B$ be a multiplicative perturbation of $A\in\mathbb{C}^{m\times n}$ given by $B=D_1^* A D_2$, where $D_1\in\mathbb{C}^{m\times m}$ and $D_2\in\mathbb{C}^{n\times n}$ are both nonsingular. New upper bounds for $\Vert B^\dag-A^\dag\Vert_U$ and $\Vert B^\dag-A^\dag\Vert_Q$ are derived, where $A^\dag,B^\dag$ are the Moore-Penrose inverses of $A$ and $B$, and $\Vert \cdot\Vert_U,\Vert \cdot\Vert_Q$ are any unitarily invariant norm and $Q$-norm, respectively. Numerical examples are provided to illustrate the sharpness of the obtained upper bounds.
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