Section 3.5: Eigenvalues and eigenvectors of a symmetric matrix

Symbolic eigenvalues and eigenvectors

To obtain symbolic (exact) eigenvalues and eigenvectors, it is only necessary to define the matrix to be symbolic:

A=sym([1,2,-1;4,0,1;-7,-2,3])

A =

[ 1, 2, -1]

[ -7, -2, 3]

V =

[ - (11*15^(1/2))/34 - 43/34, (11*15^(1/2))/34 - 43/34, 1/2]

[ 1, 1, 1]

D =

[ 0, 15^(1/2) + 1, 0]

[ 0, 0, 2]
Recall that you can use the pretty command to make output easier to read. For example, here is the second eigenvector:
pretty(V(:,2))

+- -+

| 4 15 |

| 11/17 - ------- |

| 17 |

| 1/2 |

| -------- - 43/34 |

| 34 |

| |

+- -+
It is not always possible to compute eigenvalues and eigenvectors exactly. Indeed, the eigenvalues of an n by n matrix are the roots of the nth-degree characteristic polynomial. One of the most famous results of 19^th century mathematics is that it is impossible to find a formula (analogous to the quadratic formula) expressing the roots of an arbitrary polynomial of degree 5 or more. For this reason, MATLAB cannot always find the eigenvalues of a symbolic matrix exactly. When it cannot, it automatically computes them approximately, using high precision arithmetic.

H=sym(hilb(5))

H =

[ 1, 1/2, 1/3, 1/4, 1/5]

[ 1/3, 1/4, 1/5, 1/6, 1/7]

[ 1/4, 1/5, 1/6, 1/7, 1/8]

[ 1/5, 1/6, 1/7, 1/8, 1/9]

ans =

-0.31015050746487107774850800716253

1.3453013982367055055233302583319

-2.0390705016861528742291328798663

1.0
Notice that the result appears to be in floating point, but with a large number of digits. In fact, MATLAB has returned a symbolic quantity, as the command whos shows:
whos

Name Size Bytes Class Attributes

H 5x5 60 sym

V 5x5 60 sym

ans 5x1 60 sym

ans =

2.5796608527140381819915257178503*10^(-40)

-3.2318146482723256106713975379415*10^(-41)

1.8942051991510714756211514642121*10^(-40)

8.1080530444298240540717917418738*10^(-41)
The explanation of these results is that when MATLAB could not compute the eigenpairs symbolically, it automatically switched to variable precision arithmetic, which, by default, uses 32 digits in all calculations. I will not have any need for variable precision arithmetic in this tutorial, but I wanted to mention it in case you encounter it unexpectedly, as in the above example. You may wish to explore this capability of MATLAB further for your own personal use. You can start with "help vpa".
Since it is not usually possible to find eigenpairs symbolically (except for small matrices), it is typical to perform a floating point computation from the very beginning.

Example 3.49

I will now use the spectral method to solve Ax=b, where A and b are as defined below:

A=[11,-4,-1;-4,14,-4;-1,-4,11]

A =

11 -4 -1

-1 -4 11

b=[1;2;1]

b =

2

1

V =

0.5774 -0.0000 0.8165

0.5774 -0.7071 -0.4082

D =

0 12.0000 0

0 0 18.0000
The solution of Ax=b is then
x=(V(:,1)'*b/D(1,1))*V(:,1)+(V(:,2)'*b/D(2,2))*V(:,2)+(V(:,3)'*b/D(3,3))*V(:,3)

x =

0.2593

Check:
A*x-b

ans =

1.0e-015 *

0.8882

Review: Functions in MATLAB

I have presented three ways to define new functions in MATLAB. I will now review and compare these three mechanisms.

First of all, an expression in one or more variables can be used to represent a function. For example, to work with the function f(x)=x², I define
clear

syms x

f=x^2

f =

x^2
Now I can do whatever I wish with this function, including evaluate it, differentiate it, and integrate it. To evaluate f, I use the subs command:
subs(f,x,3)

ans =

9
Differentiation and integration are performed with diff and int, respectively:
diff(f,x)

ans =

2*x
int(f,x)

ans =

x^3/3
As I have shown in earlier examples, a symbolic expression can be evaluated at a vector argument; that is, it is automatically vectorized:
t=linspace(0,1,21);
plot(t,subs(f,x,t))


The second way to define a function is using the @ operator:
clear

f=@(x)x^2

f =

@(x)x^2
An advantage of using a function is that functional notation is used for evaluation (instead of the subs command).

f(3)

ans =

9
Functions can be evaluated with symbolic inputs:
syms a

f(a)

ans =

a^2
Symbolic calculus operations can be performed on functions indirectly, bearing in mind that diff and int operate on expressions, not functions.

syms x

diff(f(x),x)

ans =

2*x
Here is an important fact to notice: functions are not automatically vectorized:

t=linspace(0,1,41);

plot(t,f(t))

??? Error using ==> mpower

Inputs must be a scalar and a square matrix.

To compute elementwise POWER, use POWER (.^) instead.
Error in ==> @(x)x^2
You can vectorize a function when you write it by using ".^", ".*", and "./".
g=@(x)x.^2;

plot(t,g(t))

Here is a common situation that I often face: Suppose we compute a complicated symbolic expression, perhaps using diff or another command, and we wish to make it into a function without retyping it. (We might wish to do this inside a script, for example, where it is not possible to retype the expression, or copy and paste it, because the calculations are being done automatically by the code.) Here is a trick: convert the expression to a string using the char command, construct a string containing the command to define the function, and then execute it using the eval command.

clear

syms x y

u=x^2*(1-x)*sin(pi*y)^2

u =

-x^2*sin(pi*y)^2*(x - 1)

expr=-diff(u,x,2)-diff(u,y,2)

expr =

2*sin(pi*y)^2*(x - 1) + 4*x*sin(pi*y)^2 + 2*pi^2*x^2*cos(pi*y)^2*(x - 1) - 2*pi^2*x^2*sin(pi*y)^2*(x - 1)

cmd=['f=@(x,y)',char(expr)]

cmd =

f=@(x,y)2*sin(pi*y)^2*(x - 1) + 4*x*sin(pi*y)^2 + 2*pi^2*x^2*cos(pi*y)^2*(x - 1) - 2*pi^2*x^2*sin(pi*y)^2*(x - 1)
eval(cmd)

f =

@(x,y)2*sin(pi*y)^2*(x-1)+4*x*sin(pi*y)^2+2*pi^2*x^2*cos(pi*y)^2*(x-1)-2*pi^2*x^2*sin(pi*y)^2*(x-1)
f(.5,.75)

ans =

0.5000
In the above calculation, notice that "['f=@(x,y)',char(expr)]" creates a character array---a string---that is the result of concatenating the two strings 'f=@(x,y)' and char(expr). Also note that single quotes are used to delimit an explicit string in MATLAB.
One problem with the previous example is that the resulting function f is not vectorized. This is easily remedied using the MATLAB command vectorize:
clear

syms x y

u=x^2*(1-x)*sin(pi*y)^2;

expr=-diff(u,x,2)-diff(u,y,2);

cmd=['f=@(x,y)',vectorize(char(expr))]

cmd =

f=@(x,y)2.*sin(pi.*y).^2.*(x - 1) + 4.*x.*sin(pi.*y).^2 + 2.*pi.^2.*x.^2.*cos(pi.*y).^2.*(x - 1) - 2.*pi.^2.*x.^2.*sin(pi.*y).^2.*(x - 1)

The third way to define a new function in MATLAB is to write an M-file. The major advantage of this method is that very complicated functions can be defined, in particular, functions that require several MATLAB commands to evaluate. You can review the section "Programming in MATLAB, part I" at this point, if necessary.

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