The derivative of a matrix element with respect to the matrix:

Is there any equation or formula for it?

I could not find it on the Web. If you know any book treating it, I would appreciate the title.

Thanks!

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- Feb 26th 2010, 02:03 PMpaolopiaceDerivative of matrix element
The derivative of a matrix element with respect to the matrix:

Is there any equation or formula for it?

I could not find it on the Web. If you know any book treating it, I would appreciate the title.

Thanks! - Feb 27th 2010, 07:22 AMHallsofIvy
Derivatives only make sense for

**functions**so this matrix and matrix element must have one or more variables. Let's say that each element, and so the matrix, if a function of t. Then is the derivative of the function and is the matrix having those derivatives as elements.

By the chain rule, .

If that inverse does not exist, the derivative does not exist. - Feb 27th 2010, 09:04 AMpaolopiace
- Feb 27th 2010, 09:39 AMOpalg
I agree with HallsofIvy. It is not orthodox mathematics to differentiate with respect to a matrix. Nevertheless, some people have tried to formulate such a concept, and you may find this Wikipedia page informative.

According to that page, if f(X) is a scalar-valued function of an n×m matrix X then the derivative is defined to be the m×n matrix whose (i,j)-element is . In particular, if then would be a matrix with a 1 in the (j,i)-position and zeros elsewhere.

But notice that much of the material on that Wikipedia page is disputed. I have no idea how reliable or useful this whole concept is. - Feb 27th 2010, 10:05 AMpaolopiace
Oplag, Thanks.

I see it's better if I post a shorter version of the whole function.

I need to obtain the analytic formula of following derivative:

where the dxd matrix Sigma is positive definite and decomposed as

is the n-dimensional vector composed of the diagonal of the matrix Sigma.

Although not much relevant, where ai is the i-th row of A.

Thanks for any help.

P.S. Actually, like it could be that . Thus, when i=j. Zero when i<>j.