Re: Determinant proof
- From: rob@xxxxxxxxxxxxxx (Rob Johnson)
- Date: Sat, 10 Nov 2007 10:13:48 GMT
In article <y8zsl3e3nuy.fsf@xxxxxxxxxxxxxxxxxxxx>,
Bill Dubuque <wgd@xxxxxxxxxxxxxxxxxxxx> wrote:
istvankaster@xxxxxxxxx wrote:
On nov. 9, 11:19, ptresadern@xxxxxxxxxxxxxx wrote:
I have three matrices (in Matlab notation):
a = rand(4,4); b = rand(3,3); p = rand(3,4);
I'm trying to find an analytical proof of the following statement:
det(eye(size(b))+b*p*inv(a)*p') = det(eye(size(a))+inv(a)*p'*b*p)
http://en.wikipedia.org/wiki/Sylvester%27s_determinant_theorem
I.e. det(I+AB) = det(I+BA). I don't ever recall seeing this called
"Sylvester's determinant theorem". Does anyone know a reference?
The proof is easy: in the polynomial ring Z[Aij,Bij] simply
take the det of (I+AB)A = A(I+BA), and then cancel det(A)
What is det(A) when A is non-square?
See my prior posts [1] for much more on such "universal" proofs.
Another proof arises from the following Schur decomposition,
but that is far too much effort compared to the above proof.
[ I A ] [ I 0 ] [ I 0 ] [ I A ]
[ ] = [ ] [ ] [ ]
[ B I ] [ B I ] [ 0 I-BA ] [ 0 I ]
[ I A ] [ I-AB 0 ] [ I 0 ]
= [ ] [ ] [ ]
[ 0 I ] [ 0 I ] [ B I ]
It is not too much effort if it works, which it does, whether A is
square or not. It is quite similar to the proof in the paper on
JSTOR cited on Wikipedia: <http://tinyurl.com/37zxgd>.
<http://en.wikipedia.org/wiki/Sylvester's_determinant_theorem>
Rob Johnson <rob@xxxxxxxxxxxxxx>
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