It's used in a certain proof that I'm reading. A is a linear map from a vectorspace V onto itself.(adsbygoogle = window.adsbygoogle || []).push({});

They say they can rewrite the vector space as [itex]\mathcal V = \bigoplus_\mu \mathbb C^{m_\mu} \otimes \mathcal V^\mu[/itex] and I understand this, but they then claim one can (always, as any linear map) rewrite A as [itex]A = [A^{\mu \nu}]_{\mu \nu}[/itex] "where [itex]A^{\mu \nu}[/itex] is a linear map of [itex]\mathbb C^{m_\nu} \otimes \mathcal V^\nu[/itex] to [itex]\mathbb C^{m_\mu} \otimes \mathcal V^\mu[/itex]."

I don't understand the nature of this decomposition/rewriting. Note that this rewriting has to be possible for any A, it doesn't use any special properties of A (that comes later in the proof).

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# What does this notation mean? Linear map A = [A^\mu \nu]_\mu \nu

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