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

Let A be a matrix and define following function exp: |R -> GL(n): t |->
Sum_i=0^oo A^i*t^i/i! (just like one would define exp(a*t) for a real a).

Let us denote this function by exp(A*t). This is called a 1 - parameter
group, since exp(A*(t+s)) = exp(A*t) * exp(A*s) (as matrices).

Then d( det(exp(A*t) ) /dt |t=0 = Trace(A).

Hint:

1) Note that if A and B are similar then so are exp(A*t) and exp(B*t).

2) Note that if J is the Jordan form of A then
det( exp(J*t) )=det( exp(A*t) ), and this equals the product
exp(a1*t)*exp(a2*t)...*exp(an*t) where a1, .., an are the eigenvalues of A
(and J).

3) Note that the derivative with respect to t at the value t=0 equals
a1+a2+...an

Conclusion: the answer to the subject line "Derivative of the Determinant
Function" is obviously: the Trace Function.



.

Relevant Pages

  • Re: Derivative of the Determinant Function
    ... Corey wrote: ... function as a function of the linear space of n x n matrices, ... I think that you are asking for the derivative of the determinant at the identity matrix, that is, what is fif: ...
    (sci.math)
  • Re: Derivative of the Determinant Function
    ... Corey wrote: ... function as a function of the linear space of n x n matrices, ... of nXn matrices has dimension n^2, and hence is equivalent to n^2 ... scalar variables. ...
    (sci.math)
  • Re: question about hat matrix X(XX)^(-1)X
    ... Suppose the set of vectors Y1,..,Yk is also a basis for linear space ... Clearly the linear space spanned by the Xs equals the linear space ... algebraic manipulation alone will give you ...
    (sci.math)
  • Re: change of basis
    ... Suppose the set of vectors Y1,..,Yk is also a basis for linear space ... Clearly the linear space spanned by the Xs equals the linear space ... Please don't post the same question multiple times in separate ...
    (sci.math)
  • Re: change of basis
    ... Suppose the set of vectors Y1,..,Yk is also a basis for linear space ... Clearly the linear space spanned by the Xs equals the linear space ... Did the earlier hint "Projection Operator" not help you? ...
    (sci.math)