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Re: Multivariate pdf of a normal distribution
From: |
Mike Miller |
Subject: |
Re: Multivariate pdf of a normal distribution |
Date: |
Sun, 6 Nov 2005 11:37:26 -0600 (CST) |
On Sun, 6 Nov 2005, Gorazd Brumen wrote:
Hello Paul,
I don't know for sure that inv(r') == inv(r)' for r upper triangular.
Numerically it is not the case in octave:
octave:34> x = triu(rand(10)); norm(inv(x') - inv(x)')
ans = 3.3466e-14
Assuming that it is, then
I think I have proven that this is the case (at least for upper
triangular matrices).
But perhaps it is the case actually for all invertible matrices, not only
for upper triangular ones.
When inv(A) exists,
inv(A') = inv(A)'
because the inverse equals the transpose of the adjoint divided by the
determinant. The adjoint of the transpose equals the transpose of the
adjoint and the determinant is the same for A and A'. The identity really
follows from the fact that the determinant is not affected by the
transpose operation -- the adjoint is a collection of determinants.
I agree that this must be in most matrix algebra books.
Mike
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- Re: Multivariate pdf of a normal distribution, (continued)
- Re: Multivariate pdf of a normal distribution, Mike Miller, 2005/11/05
- Re: Multivariate pdf of a normal distribution, Prasenjit Kapat, 2005/11/05
- Re: Multivariate pdf of a normal distribution, Mike Miller, 2005/11/05
- Re: Multivariate pdf of a normal distribution, Prasenjit Kapat, 2005/11/05
- Re: Multivariate pdf of a normal distribution, Paul Kienzle, 2005/11/05
- Re: Multivariate pdf of a normal distribution, Mike Miller, 2005/11/06
Re: Multivariate pdf of a normal distribution, Michael Creel, 2005/11/07
Re: Multivariate pdf of a normal distribution, Gorazd Brumen, 2005/11/06
- Re: Multivariate pdf of a normal distribution,
Mike Miller <=