Re: Multivariate pdf of a normal distribution

[Mike Miller]

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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