Re: Derivative of the Determinant Function

Corey wrote:
Hi there. If I were to compute the derivative of the determinant
function as a function of the linear space of n x n matrices, how would
I go about doing this? My initial thought was to treat this as a Hodge
duality problem, but I'm not really sure where to begin if I take this
approach. I was just wondering if anyone could offer me some input,
thanks.

-Corey


I think that you are asking for the derivative of the determinant at the identity matrix, that is, what is f(H) if:

det(I+hA) = 1 + h f(A) + o(h).

So, det(I+hA) = (e1+ha1)^(e2+ha2)^...^(en+han)

where e1,...,en are the unit vectors, and a1,...,an are the columns of A. Well, this is going to be

ha1^e2^...^en + e1^ha2^...^en + ... + e1^e2^...^han + O(h^2)
=
h Tr(A) e1^e2^...^en + O(h^2).

Is this what you were looking for?

.

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