Perturbation Bounds for the Polar Decomposition

Let $M_n ( F )$ denote the space of matrices over the field F. Given $A \in M_n ( F )$ define $| A | \equiv ( A^ * A )^{1/2} $ and $U( A ) \equiv A | A |^{ - 1} $ assuming A is nonsingular. Let $\sigma _1 ( A ) \geq \sigma _2 ( A ) \geq \cdots \sigma _n ( A ) \geq 0$ denote the ordered singular values of A.Majorization results are obtained relating the singular values of $U ( A + \Delta A ) - U ( A )$ and those of A and $\Delta A$. In particular, it is shown that if $A,\,\Delta A \in M_n ( R )$ and $\sigma _1 ( \Delta A ) < \sigma _n ( A )$, then for any unitarily invariant norm $\| \cdot \|, \| U ( A + \Delta A ) - U ( A ) \| \leq 2 [ \sigma_{n - 1} ( A ) + \sigma_{n} ( A )]^{ - 1} \| \Delta A \|$. Similar results are obtained for matrices with complex entries.Also considered is the unitary Procrustes problem: $\min \{ \| A - UB \|:U \in M_n ( C ),U^ * U = I \}$ where $A,B \in M_n ( C )$, and a unitarily invariant norm $\| \cdot \|$ are given. It was conjectured that if U is unitary and $U^ * BA^ * $ is ...