| Abstract |
We consider pairs of linear mappings
$(\cal A,\cal B)$ of the form
$V\underset{\overleftarrow{\mathcal{\phantom{1}B\phantom{1}}}}{\xrightarrow{\mathcal{A}}}W$
in which $V$ and $W$ are
finite dimensional unitary or Euclidean spaces over $\mathbb{C}$
or $\mathbb{R}$, respectively. Let $(\cal A,\cal B)$ be
transformed to $V\underset{\overleftarrow{\phantom{1}\mathcal{B'}\phantom{1}}}{\xrightarrow{\mathcal{A'}}}W$
by bijections
$\varphi_1\colon V\to V'$ and $\varphi_2\colon W\to W'$. We say
that $(\cal A,\cal B)$ and $(\cal A',\cal B')$ are linearly
equivalent if $\varphi_1$ and $\varphi_2$ are linear bijections
and topologically equivalent if $ \varphi_1 $ and $ \varphi_2 $
are homeomorphisms. We prove that $(\cal A,\cal B)$ and $(\cal
A',\cal B')$ are topologically equivalent if and only if their
regular parts are topologically equivalent and their singular
parts are linearly
equivalent. |
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