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# Multivariable CalculusMatrix differentiation

Czas czytania: ~10 min

Just as elementary differentiation rules are helpful for optimizing single-variable functions, matrix differentiation rules are helpful for optimizing expressions written in matrix form. This technique is used often in statistics.

Suppose is a function from to . Writing , we define the Jacobian matrix (or derivative matrix) to be

Note that if , then differentiating with respect to is the same as taking the gradient of .

With this definition, we obtain the following analogues to some basic single-variable differentiation results: if is a constant matrix, then

The third of these equations is the rule.

The Hessian of a function may be written in terms of the matrix differentiation operator as follows:

Some authors define to be , in which case the Hessian operator can be written as .

Exercise
Let be defined by where is a symmetric matrix. Find

Solution. We can apply the product rule to find that

Exercise
Suppose is an matrix and . Use matrix differentiation to find the vector which minimizes . Hint: begin by writing as . You may assume that the rank of is .

Solution. We write

To minimize this function, we find its gradient

and set it equal to to get

(We know that has an inverse matrix because its rank is equal to that of , which we assumed was .)

Bruno