Create symbolic scalar variables and functions, and matrix variables and functions (2024)

Create symbolic scalar variables x and y.

syms x yx

x =$$x$$

y

y =$$y$$

Create a 1-by-4 vector of symbolic scalar variables a with the automatically generated elements $$a_1,\dots ,a_4$$. This command also creates the symbolic scalar variables a1, ..., a4 in the MATLAB workspace.

syms a [1 4]a

a =$$\left(\begin{array}{cccc}{a}_1& a_2& a_3& a_4\end{array}\right)$$

whos

 Name Size Bytes Class Attributes a 1x4 8 sym a1 1x1 8 sym a2 1x1 8 sym a3 1x1 8 sym a4 1x1 8 sym 

You can change the naming format of the generated elements by using a format character vector. Declare the symbolic scalar variables by enclosing each variable name in single quotes. syms replaces %d in the format character vector with the index of the element to generate the element names.

syms 'p_a%d' 'p_b%d' [1 4]p_a

p_a =$$\left(\begin{array}{cccc}{p}_{\mathrm{a1}}& p_{\mathrm{a2}}& p_{\mathrm{a3}}& p_{\mathrm{a4}}\end{array}\right)$$

p_b

p_b =$$\left(\begin{array}{cccc}{p}_{\mathrm{b1}}& p_{\mathrm{b2}}& p_{\mathrm{b3}}& p_{\mathrm{b4}}\end{array}\right)$$

Create a 3-by-4 matrix of symbolic scalar variables with automatically generated elements. The elements are of the form $$A_{i,j}$$, which generates the symbolic matrix variables $$A_{1,1},\dots ,A_{3,4}$$.

syms A [3 4]A

A =
$$\left(\begin{array}{cccc}{A}_{1,1}& A_{1,2}& A_{1,3}& A_{1,4}\\ A_{2,1}& A_{2,2}& A_{2,3}& A_{2,4}\\ A_{3,1}& A_{3,2}& A_{3,3}& A_{3,4}\end{array}\right)$$

Create symbolic scalar variables x and y, and assume that they are integers.

syms x y integer

Create another scalar variable z, and assume that it has a positive rational value.

syms z positive rational

Check assumptions.

assumptions

ans =$$\left(\begin{array}{cccc}x\in \mathbb{Z}& y\in \mathbb{Z}& z\in \mathbb{Q}& 0<z\end{array}\right)$$

Alternatively, check assumptions on each variable. For example, check assumptions set on the variable x.

assumptions(x)

ans =$$x\in \mathbb{Z}$$

Clear assumptions on x, y, and z.

assume([x y z],'clear')assumptions

 ans = Empty sym: 1-by-0 

Create a 1-by-3 symbolic array a, and assume that the array elements have real values.

syms a [1 3] realassumptions

ans =$$\left(\begin{array}{ccc}{a}_1\in \mathbb{R}& a_2\in \mathbb{R}& a_3\in \mathbb{R}\end{array}\right)$$

Create symbolic functions with one and two arguments.

syms s(t) f(x,y)

Both s and f are abstract symbolic functions. They do not have symbolic expressions assigned to them, so the bodies of these functions are s(t) and f(x,y), respectively.

Specify a formula for f.

f(x,y) = x + 2*y

f(x, y) =$$x+2\hspace{0.17em}y$$

Compute the function value at the point x = 1 and y = 2.

f(1,2)

ans =$$5$$

Create a symbolic function and specify its formula by using a matrix of symbolic scalar variables.

syms xM = [x x^3; x^2 x^4];f(x) = M

f(x) =
$$\left(\begin{array}{cc}x& x^3\\ x^2& x^4\end{array}\right)$$

Compute the function value at the point x = 2.

f(2)

ans =
$$\left(\begin{array}{cc}2& 8\\ 4& 16\end{array}\right)$$

Compute the value of this function for x = [1 2 3; 4 5 6]. The result is a cell array of symbolic matrices.

xVal = [1 2 3; 4 5 6];y = f(xVal)

y=2×2 cell array {2x3 sym} {2x3 sym} {2x3 sym} {2x3 sym}

Access the contents of a cell in the cell array by using braces.

y{1}

ans =
$$\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right)$$

Create a 2-by-2 symbolic matrix with automatically generated symbolic functions as its elements.

syms f(x,y) 2f

f(x, y) =
$$\left(\begin{array}{cc}{f}_{1,1}\left(x,y\right)& f_{1,2}\left(x,y\right)\\ f_{2,1}\left(x,y\right)& f_{2,2}\left(x,y\right)\end{array}\right)$$

Assign symbolic expressions to the symbolic functions f1_1(x,y) and f2_2(x,y). These functions are displayed as $$f_{1,1}(x,y)$$ and $$f_{2,2}(x,y)$$ in the Live Editor. When you assign these expressions, the symbolic matrix f still contains the initial symbolic functions in its elements.

f1_1(x,y) = 2*x;f2_2(x,y) = x - y;f

f(x, y) =
$$\left(\begin{array}{cc}{f}_{1,1}\left(x,y\right)& f_{1,2}\left(x,y\right)\\ f_{2,1}\left(x,y\right)& f_{2,2}\left(x,y\right)\end{array}\right)$$

Substitute the expressions assigned to f1_1(x,y) and f2_2(x,y) by using the subs function.

A = subs(f)

A(x, y) =
$$\left(\begin{array}{cc}2\hspace{0.17em}x& f_{1,2}\left(x,y\right)\\ f_{2,1}\left(x,y\right)& x-y\end{array}\right)$$

Evaluate the value of the symbolic matrix A, which contains the substituted expressions, at x = 2 and y = 3.

A(2,3)

ans =
$$\left(\begin{array}{cc}4& f_{1,2}\left(2,3\right)\\ f_{2,1}\left(2,3\right)& -1\end{array}\right)$$

Since R2021a

Create two symbolic matrix variables with size 2-by-3. Nonscalar symbolic matrix variables are displayed as bold characters in the Live Editor and Command Window.

syms A B [2 3] matrixA

A =$$A$$

B

B =$$B$$

Add the two matrices. The result is represented by the matrix notation $$\text{A}+\text{B}$$.

X = A + B

X =$$A+B$$

The data type of X is symmatrix.

class(X)

ans = 'symmatrix'

Convert the symbolic matrix variable X to a matrix of symbolic scalar variables Y. The result is denoted by the sum of the matrix components.

Y = symmatrix2sym(X)

Y =
$$\left(\begin{array}{ccc}{A}_{1,1}+B_{1,1}& A_{1,2}+B_{1,2}& A_{1,3}+B_{1,3}\\ A_{2,1}+B_{2,1}& A_{2,2}+B_{2,2}& A_{2,3}+B_{2,3}\end{array}\right)$$

The data type of Y is sym.

class(Y)

ans = 'sym'

Show that the converted result in Y is equal to the sum of two matrices of symbolic scalar variables.

syms A B [2 3]Y2 = A + B

Y2 =
$$\left(\begin{array}{ccc}{A}_{1,1}+B_{1,1}& A_{1,2}+B_{1,2}& A_{1,3}+B_{1,3}\\ A_{2,1}+B_{2,1}& A_{2,2}+B_{2,2}& A_{2,3}+B_{2,3}\end{array}\right)$$

isequal(Y,Y2)

ans = logical 1

Since R2021a

Symbolic matrix variables represent matrices, vectors, and scalars in compact matrix notation. When representing nonscalars, these variables are noncommutative. When mathematical formulas involve matrices and vectors, writing them using symbolic matrix variables is more concise and clear than writing them componentwise.

Create two symbolic matrix variables.

syms A B [2 2] matrix

Check the commutation relation for multiplication between two symbolic matrix variables.

A*B - B*A

ans =$$A\hspace{0.17em}B-B\hspace{0.17em}A$$

isequal(A*B,B*A)

ans = logical 0

Check the commutation relation for addition between two symbolic matrix variables.

isequal(A+B,B+A)

ans = logical 1

Since R2021a

Create 3-by-3 and 3-by-1 symbolic matrix variables.

syms A 3 matrixsyms X [3 1] matrix

Find the Hessian matrix of $${\text{X}}^T\text{A}\text{X}$$. Derived equations involving symbolic matrix variables are displayed in typeset as they would be in textbooks.

f = X.'*A*X

f =$$X^{\mathrm{T}}\hspace{0.17em}A\hspace{0.17em}X$$

H = diff(f,X,X.')

H =$$A^{\mathrm{T}}+A$$

Since R2022a

Create the function $\mathit{v}\left(\mathit{r},\mathit{t}\right)=\frac{{\Vert \mathit{r}\Vert }_2}{\mathit{t}}$, where $\mathit{r}$ is a 1-by-3 vector and $\mathit{t}$ is a scalar.

Create $\mathit{r}$ as a symbolic matrix variable and $\mathit{t}$ as a symbolic scalar variable. Create $\mathit{v}\left(\mathit{r},\mathit{t}\right)$ as a symbolic matrix function and keep existing definitions of $\mathit{r}$ and $\mathit{t}$ in the workspace.

syms r [1 3] matrixsyms tsyms v(r,t) [1 1] matrix keepargs

Assign the formula of $\mathit{v}\left(\mathit{r},\mathit{t}\right)$.

v(r,t) = norm(r)/t

v(r, t) =$${\Vert r\Vert }_2\hspace{0.17em}{t}^{-1}$$

The symbolic matrix function accepts scalars, vectors, and matrices as its input arguments. Evaluate the function for the vector value $\mathit{r}=\left(1,2,2\right)$ and the scalar value $\mathit{t}=3$.

vEval = v([1 2 2],3)

vEval =
$$\begin{array}{l}\frac{1}{3}\hspace{0.17em}{\Vert {\Sigma }_1\Vert }_2\\ \\ \mathrm{where}\\ \\ \mathrm{}{\Sigma }_1=\left(\begin{array}{ccc}1& 2& 2\end{array}\right)\end{array}$$

Convert the evaluated function from the symmatrix data type to the sym data type.

vSym = symmatrix2sym(vEval)

vSym =$$1$$

Certain functions, such as solve and symReadSSCVariables, can return a vector of symbolic scalar variables or a cell array of symbolic scalar variables and functions. These variables or functions do not automatically appear in the MATLAB workspace. Create these variables or functions from the vector or cell array by using syms.

Solve the equation sin(x) == 1 by using solve. The parameter k in the solution does not appear in the MATLAB workspace.

syms xeqn = sin(x) == 1;[sol,parameter,condition] = solve(eqn,x,'ReturnConditions',true);parameter

parameter =$$k$$

Create the parameter k by using syms. The parameter k now appears in the MATLAB workspace.

syms(parameter)

Similarly, use syms to create the symbolic objects contained in a vector or cell array. Examples of functions that return a cell array of symbolic objects are symReadSSCVariables and symReadSSCParameters.

Create some symbolic scalar variables, functions, matrix variables, matrix functions, and arrays.

syms a f(x)syms A [2 1]syms B [2 2] matrixsyms g(B) [2 2] matrix keepargs

Display a list of all symbolic scalar variables, functions, matrix variables, matrix functions, and arrays that currently exist in the MATLAB workspace by using syms.

syms

Your symbolic variables are:A A1 A2 B a f g x 

Instead of displaying a list, return a cell array by providing an output to syms.

S = syms

S = 8x1 cell {'A' } {'A1'} {'A2'} {'B' } {'a' } {'f' } {'g' } {'x' }

Create several symbolic objects.

syms a b c f(x)syms D g(y) [2 2] matrix

Return all symbolic objects as a cell array by using the syms function. Use the cellfun function to delete all symbolic objects in the cell array symObj.

symObj = syms;cellfun(@clear,symObj)

Check that you deleted all symbolic objects by calling syms. The output is empty, meaning no symbolic objects exist in the MATLAB workspace.

syms

S — Names of all symbolic scalar variables, functions, matrix variables, matrix functions, and arrays
cell array of character vectors

Names of all symbolic scalar variables, functions, matrix variables, matrix functions, and arrays in the MATLAB workspace, returned as a cell array of character vectors.