Help! Changing of variables... Jacobian needed...

Hi all,

Suppose I have a vector Z=(Z1, Z2, Z3... Zn) in the column form,

I am trying to evaluate the multi-variate integral:

Integrate(C*(Z'*K^(-1)*Z)*exp(-1/2*(Z'*K^(-1)*Z)), for dZ=(dZ1, dZ2, ...,
dZn))

where K is a given matrix and Z' is a transpose of Z column vector.

I am trying hard to do changing of variables: I know (Z'*K^(-1)*Z) is a
single value, 1x1, so I want to define a new variable y=Z'*K^(-1)*Z...

So the above integral changes into:

Integrate(C*y*exp(-1/2*y)*SOME_THING, for dy)

but I am trying hard to figure out that "SOME_THING",

y=Z'*K^(-1)*Z, so dy=K^(-1)*Z*dZ, but dZ does not equal to dy/(K^(-1)*Z),
since dy is a single variable, and K^(-1)*Z is a vector... so this is not
valid.

Then I have to do Jacobian. But the Jacobian won't be square since there is
only 1 y vs. so many Z's.

I can define dummy vector W=(Z'*K^(-1)*Z, Z2, Z3, Z4, ..., Zn),

then Jacobian(W/Z)=| VERY_CHAOTIC_TERMS | which is beyond my brain ...

Can anybody show me how to do this Jacobian? or do you have a better
Jacobian?

Thanks a lot!

-L





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Relevant Pages

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  • Jacobian Pseudoinverse
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