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Symbolic Toolbox

In addition to doing numeric computations, it is possible to do symbolic calculations with MATLAB’s symbolic toolbox. The symbolic toolbox allows us to manipulate symbolic expressions. It can be used to solve equations, differentiate and integrate functions, and perform symbolic matrix operations.

MATLAB’s sym command can be used to turn any MATLAB data structure into a symbolic object. For example, the command sym(‘t’) will turn the string ‘t’ into a symbolic variable t, and the command sym(hilb(3)) will produce the symbolic version of the $3 \times 3$ $3 \times 3$ Hilbert matrix written in the form

\begin{matrix} [\begin{matrix} 1, & \frac{1}{2}, & \frac{1}{3} \end{matrix}] \\ [\begin{matrix} \frac{1}{2}, & \frac{1}{3}, & \frac{1}{4} \end{matrix}] \\ [\begin{matrix} \frac{1}{3}, & \frac{1}{4}, & \frac{1}{5} \end{matrix}] \end{matrix}

$\begin{matrix} [\begin{matrix} 1, & \frac{1}{2}, & \frac{1}{3} \end{matrix}] \\ [\begin{matrix} \frac{1}{2}, & \frac{1}{3}, & \frac{1}{4} \end{matrix}] \\ [\begin{matrix} \frac{1}{3}, & \frac{1}{4}, & \frac{1}{5} \end{matrix}] \end{matrix}$

We can create a number of symbolic variables at once with the syms command. For example, the command

\begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} syms \end{matrix} & a \end{matrix} & b \end{matrix} & c \end{matrix}

$\begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} syms \end{matrix} & a \end{matrix} & b \end{matrix} & c \end{matrix}$

creates three symbolic variables a, b, and c. If we then set

A = [\begin{matrix} a, & b, & c; b, & c, & a; c, & a, & b \end{matrix}]

$A = [\begin{matrix} a, & b, & c; b, & c, & a; c, & a, & b \end{matrix}]$

the result will be the symbolic matrix

\begin{array}{l} A = \\ [\begin{matrix} \begin{matrix} a, & b, \end{matrix} & c \end{matrix}] \\ [\begin{matrix} \begin{matrix} b, & c, \end{matrix} & a \end{matrix}] \\ [\begin{matrix} \begin{matrix} c, & a, \end{matrix} & b \end{matrix}] \end{array}

$\begin{array}{l} A = \\ [\begin{matrix} \begin{matrix} a, & b, \end{matrix} & c \end{matrix}] \\ [\begin{matrix} \begin{matrix} b, & c, \end{matrix} & a \end{matrix}] \\ [\begin{matrix} \begin{matrix} c, & a, \end{matrix} & b \end{matrix}] \end{array}$

The MATLAB command subs can be used to substitute an expression or a value for a symbolic variable. For example, the command subs(A, c, 3) will substitute 3 for each occurrence of c in the symbolic matrix A. Multiple substitutions are also possible: The command

subs (A, [a, b, c], [a - 1, b+1, 3])

$subs (A, [a, b, c], [a - 1, b+1, 3])$

will substitute $a - 1, b + 1$ $a - 1, b + 1$ , and 3 for a, b, and c, respectively, in the matrix A.

The standard matrix operations ∗,ˆ, +, −, and ′ all work for symbolic matrices and also for combinations of symbolic and numeric matrices. If an operation involves two matrices and one of them is symbolic, the result will be a symbolic matrix. For example, the command

sym (hilb (3)) + eye (3)

$sym (hilb (3)) + eye (3)$

will produce the symbolic matrix

\begin{matrix} [\begin{matrix} \begin{matrix} 2, & \frac{1}{2}, \end{matrix} & \frac{1}{3} \end{matrix}] \\ [\begin{matrix} \begin{matrix} \frac{1}{2}, & \frac{4}{3}, \end{matrix} & \frac{1}{4} \end{matrix}] \\ [\begin{matrix} \begin{matrix} \frac{1}{3}, & \begin{matrix} \frac{1}{4}, & \frac{6}{5} \end{matrix} \end{matrix} \end{matrix}] \end{matrix}

$\begin{matrix} [\begin{matrix} \begin{matrix} 2, & \frac{1}{2}, \end{matrix} & \frac{1}{3} \end{matrix}] \\ [\begin{matrix} \begin{matrix} \frac{1}{2}, & \frac{4}{3}, \end{matrix} & \frac{1}{4} \end{matrix}] \\ [\begin{matrix} \begin{matrix} \frac{1}{3}, & \begin{matrix} \frac{1}{4}, & \frac{6}{5} \end{matrix} \end{matrix} \end{matrix}] \end{matrix}$

Standard MATLAB matrix commands such as

det, eig, inv, null, trace, sum, prod, poly

$det, eig, inv, null, trace, sum, prod, poly$

all work for symbolic matrices; however, others such as

rref, orth, rank, norm

$rref, orth, rank, norm$

do not. Likewise, none of the standard matrix factorizations are possible for symbolic matrices.

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Symbolic Toolbox