Partial Derivatives of Matrix/Vector Function: An Easier Way?

[datahead8888]

I was working on a pde, and I needed to compute a Jacobian for it.

Suppose we have a function consisting of a series of matrices multiplied by a vector:
f(X) = A * B * b
--where X is a vector containing elements that are contained within A, b, and/or b,
--A is a matrix, B is a matrix, and b is a vector

Each Matrix and the vector is expressed as more terms, ie...
X = (x1, x2, x3)

A =
[ x1 + y1 y4 y7 ]
[ y2 x2 + y5 y8 ]
] y3 y6 x3 + y9 ]

B =
[ y1 x2 + y4 x3 + y7 ]
[x1 + y2 y5 y8 ]
] y3 y6 y9 ]

b = [y1 y2 y3]' (' means transposed)

Now we want to find the Jacobian of f - ie the partial derivative of f wrt X.

One way to do this is to multiply the two matrices and then multiply that by the vector, creating one 3x1 vector in which each element is an algebraic expression resulting from matrix multiplication. The partial derivative could then be computed per element to form a 3x3 Jacobian. This would be feasible in the above example, but the one I'm working is a lot more complicated (and so I would also have to look for patterns in order to simplify it afterwards).

I was wanting to try to use the chain rule and/or the product rule for partial derivatives if possible. However, with the product rule you end up with A' * B * b + A * B' * b + A * B * b', where each derivative is wrt to the vector X. I understand that the derivative of a matrix wrt a vector is actually a 3rd order tensor, which is not easy to deal with. If this is not correct, the other terms still have to evaluate to matrices in order for matrix addition to be valid. If I use the chain rule instead, I still end up with the derivative of a matrix wrt a vector.

Is there an easier way to break down a matrix calculus problem like this? I've scoured the web and cannot seem to find a good direction.

 

[I like Serena]

I was working on a pde, and I needed to compute a Jacobian for it.

Suppose we have a function consisting of a series of matrices multiplied by a vector:
f(X) = A * B * b
--where X is a vector containing elements that are contained within A, b, and/or b,
--A is a matrix, B is a matrix, and b is a vector

Each Matrix and the vector is expressed as more terms, ie...
X = (x1, x2, x3)

A =
[ x1 + y1 y4 y7 ]
[ y2 x2 + y5 y8 ]
] y3 y6 x3 + y9 ]

B =
[ y1 x2 + y4 x3 + y7 ]
[x1 + y2 y5 y8 ]
] y3 y6 y9 ]

b = [y1 y2 y3]' (' means transposed)

Now we want to find the Jacobian of f - ie the partial derivative of f wrt X.

One way to do this is to multiply the two matrices and then multiply that by the vector, creating one 3x1 vector in which each element is an algebraic expression resulting from matrix multiplication. The partial derivative could then be computed per element to form a 3x3 Jacobian. This would be feasible in the above example, but the one I'm working is a lot more complicated (and so I would also have to look for patterns in order to simplify it afterwards).

I was wanting to try to use the chain rule and/or the product rule for partial derivatives if possible. However, with the product rule you end up with A' * B * b + A * B' * b + A * B * b', where each derivative is wrt to the vector X. I understand that the derivative of a matrix wrt a vector is actually a 3rd order tensor, which is not easy to deal with. If this is not correct, the other terms still have to evaluate to matrices in order for matrix addition to be valid. If I use the chain rule instead, I still end up with the derivative of a matrix wrt a vector.

Is there an easier way to break down a matrix calculus problem like this? I've scoured the web and cannot seem to find a good direction.

Hi datahead8888! :)

Your analysis is flawless.
Both methods would work perfectly.

Don't worry about 3rd order tensors too much though.
If it seems complicated, just start with the derivative with respect to the first variable.
No tensor in sight.
Then go on with the 2nd variable etcetera and... there you go! ;)