Integral of Log det(1-A(x).B(x)) wrt x

[guerom00]

Hello all :)

I have two square matrices whose elements are functions of a variable x, let's call them A(x) and B(x).
Those two matrices do not commute : A(x).B(x)≠B(x).A(x)
I then define the quantity Log det(1-A(x).B(x)) where 1 is the identity matrix.

I'm interested in a closed form for the integral of the above quantity wrt x i.e.
\int\,Log\,det(1-A(x).B(x))\,dx

Do you think such a closed form exists ?

Thanks in advance :)

 

[Hurkyl]

A closed form doesn't exist in the case of 1x1 matrices!

 

[guerom00]

True. So let me be more specific because, in my case, how my matrices depend on x is known

In reality I have a product of 4 matrices :
R1.Exp[-k x].R2.Exp[-k x]
Let me explain each terms :
• R1 and R2 do not depend on x but are not symmetric (hence the no commutation in all those matrices products)
• Exp[-k x] is a diagonal matrix whose elements are exp(-k x) with k a column vector.

So with these additional informations, I'm interested in a possible closed form for the quantity

\int\,Log\, det(1-R1.Exp[-k x].R2.Exp[-k x])\, dx

 

[Hurkyl]

I was about to repeat myself, but it turns out there is a closed form for 1x1 matrices

http://integrals.wolfram.com/index.jsp?expr=Log[1+-+r1+Exp[-k+x]+r2+Exp[-k+x]]&random=false

if you allow polylogs in closed forms.

 

[guerom00]

Indeed there is a closed form for numbers (1x1 matrices) which necessarily commute. I'm wondering if that can be somehow generalized for non commuting matrices quantities.
I know that if a closed form exists, it will indeed involve the dilogarithm of a matrix I'll think of what this is later… One problem at a time