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 Post subject: Invertible matricies are the exponential of another matrixPosted: Thu, 1 Dec 2011 05:25:06 UTC
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Joined: Fri, 3 Sep 2010 09:29:45 UTC
Posts: 140
Another one that I am guessing is fairly easy

Every invertible n x n matrix A can be written as for some .

The hint is to start with where is a block diagonal matrix in which each block is of the form and is nilpotent.

I know that if is unipotent then is nilpotent and if is nilpotent then is unipotent

So certainly exists, and is unipotent.

I'm not really sure where to go with this one.

Any hints?

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 Post subject: Re: Invertible matricies are the exponential of another matrPosted: Thu, 1 Dec 2011 06:07:59 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6008
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
qwirk wrote:
Another one that I am guessing is fairly easy

Every invertible n x n matrix A can be written as for some .

The hint is to start with where is a block diagonal matrix in which each block is of the form and is nilpotent.

I know that if is unipotent then is nilpotent and if is nilpotent then is unipotent

So certainly exists, and is unipotent.

I'm not really sure where to go with this one.

Any hints?

No, you want to construct your nilpotent from a unipotent matrix, not starting with N nilpotent and show is unipotent.

Recall:
• when .
• is (for those A such that the series is defined) a right inverse of .

So conjugating to decompose into (irreducible) invariant subspaces of , and obviously A|subspace commutes with A|{another subspace}. So you are down to constructing log when it is of the form , nilpotent N, by upper-triangularising (e.g. Jordan normal form).

The is easy to get rid of --- just use since it commutes with everything, so you are down to taking log of , nilpotent . That's where point (3) comes in.

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 Post subject: Re: Invertible matricies are the exponential of another matrPosted: Thu, 1 Dec 2011 08:24:33 UTC
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Joined: Fri, 3 Sep 2010 09:29:45 UTC
Posts: 140
Thanks again outermeasure

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