Change Language

English

Sign in to Mathigon

or

New Account     Reset Password     

Multivariable Calculus
by Samuel S. Watson
Introduction
Taylor Series
Differentiation
Optimization
Matrix differentiation
The chain rule
Multivariable integration
Custom integration

Share

Reset Progress

This will delete your progress and chat data for all chapters in this course, and cannot be undone!

Glossary

Select one of the keywords on the left…

Multivariable CalculusMatrix differentiation

Reading time: ~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 \mathbf{f} is a function from \mathbb{R}^n to \mathbb{R}^m. Writing \mathbf{f}(\mathbf{x}) = \mathbf{f}(x_1, \ldots, x_n), we define the Jacobian matrix (or derivative matrix) to be

\begin{align*}\frac{\partial \mathbf{f}}{\partial \mathbf{x}} = \left[ \begin{array}{cccc} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \cdots & \frac{\partial f_1}{\partial x_n} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \cdots & \frac{\partial f_2}{\partial x_n} \\ \vdots & & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1} & \frac{\partial f_m}{\partial x_2} & \cdots & \frac{\partial f_m}{\partial x_n} \end{array}\right]\end{align*}

Note that if m=1, then differentiating f with respect to \mathbf{x} is the same as taking the gradient of f.

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

\begin{align*}\frac{\partial}{\partial \mathbf{x}} (A \mathbf{x}) &= A \\\ \frac{\partial}{\partial \mathbf{x}} (\mathbf{x}' A) &= A' \\\ \frac{\partial}{\partial \mathbf{x}} (\mathbf{u}' \mathbf{v}) &= \mathbf{u}'\frac{\partial \mathbf{v}}{\partial \mathbf{x}} + \mathbf{v}'\frac{\partial \mathbf{u}}{\partial \mathbf{x}}\end{align*}

The third of these equations is the rule.

The Hessian of a function f:\mathbb{R}^n \to \mathbb{R} may be written in terms of the matrix differentiation operator as follows:

\begin{align*}\mathbf{H}(\mathbf{x}) = \frac{\partial}{\partial \mathbf{x}} \left(\frac{\partial f}{\partial \mathbf{x}}\right)'.\end{align*}

Some authors define \frac{\partial f}{\partial \mathbf{x}'} to be \left(\frac{\partial f}{\partial \mathbf{x}}\right)', in which case the Hessian operator can be written as \frac{\partial^2}{\partial \mathbf{x} \partial \mathbf{x}'}.

Exercise
Let f: \mathbb{R}^n \to \mathbb{R} be defined by f(\mathbf{x}) = \mathbf{x}' A \mathbf{x} where A is a symmetric matrix. Find \frac{\partial f}{\partial \mathbf{x}}.

Solution. We can apply the product rule to find that

\begin{align*}\frac{\partial f}{\partial \mathbf{x}} = \mathbf{x}' \frac{\partial}{\partial \mathbf{x}}(A\mathbf{x}) + (A\mathbf{x})' \frac{\partial \mathbf{x}}{\partial \mathbf{x}} = \mathbf{x}' A + \mathbf{x}' A = 2\mathbf{x}' A.\end{align*}

Exercise
Suppose A is an m\times n matrix and \mathbf{b} \in \mathbb{R}^m. Use matrix differentiation to find the vector \mathbf{x} which minimizes |A \mathbf{x} - \mathbf{b}|^2. Hint: begin by writing |A \mathbf{x} - \mathbf{b}|^2 as (A \mathbf{x} - \mathbf{b})' (A \mathbf{x} - \mathbf{b}). You may assume that the rank of A is n.

Solution. We write

\begin{align*}|A \mathbf{x} - \mathbf{b}|^2 &= (A \mathbf{x} - \mathbf{b})' (A \mathbf{x} - \mathbf{b}) \\\ &= \mathbf{x}' A' A \mathbf{x} - \mathbf{b}' A \mathbf{x} + \mathbf{x}' A' \mathbf{b} + |\mathbf{b}|^2.\end{align*}

To minimize this function, we find its gradient

\begin{align*}\frac{\partial}{\partial \mathbf{x}}|A \mathbf{x} - \mathbf{b}|^2 = 2\,\mathbf{x}' A' A - \mathbf{b}' A + (A'\mathbf{b})' = 2\mathbf{x}' A' A- 2\mathbf{b}' A\end{align*}

and set it equal to \boldsymbol{0} to get

\begin{align*}\mathbf{x}' = \mathbf{b}' A(A' A)^{-1} \implies \mathbf{x} = (A' A)^{-1} A' \mathbf{x}.\end{align*}

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

Bruno
Bruno Bruno
Popular Courses
Intermediate – Circles and Pi
Spheres, Cones and Cylinders
Intermediate – Sequences and Patterns
Fibonacci Numbers
Intermediate – Graphs and Networks
The Bridges of Königsberg
Advanced – Fractals
The Mandelbrot Set