8 7.28, §8 Comparison Of SOV and Eigenvalue Problems

The similarity between eigenvalue problems and Sturm-Liouville is not a coincidence. This section explores it a bit deeper:

8.1 Functions as vectors

First I want to convince you that there is no big difference between vectors and functions. A vector $\vec f$ is usually shown in the form of an arrow:

Figure 3: The classical picture of a vector.

However, the same vector may instead be represented as a spike diagram, by plotting the value of the components versus the component index:

Figure 4: Spike diagram of a vector.

In the same way as in two dimensions, a vector in three dimensions, or, for that matter, in thirty dimensions, can be represented by a spike diagram:

Figure 5: More dimensions.

For a large number of dimensions, and in particular in the limit of infinitely many dimensions, the large values of $i$ can be rescaled into a continuous coordinate, call it $x$. For example, $x$ might be defined as $i$ divided by the number of dimensions. In any case, the spike diagram becomes a function $f(x)$:

Figure 6: Infinite dimensions.

The spikes are usually not shown:

Figure 7: The classical picture of a function.

In this way, a function is just a vector in infinitely many dimensions.

8.2 The dot product

The dot product of vectors is an important tool. It makes it possible to find the length of a vector, by multiplying the vector by itself and taking the square root. It is also used to check if two vectors are orthogonal: if their dot product is zero, they are. In this subsection, the dot product is defined for functions.

The usual dot product of two vectors $\vec f$ and $\vec g$ can be found by multiplying components with the same index $i$ together and summing that:

$$\vec f \cdot \vec g \equiv f_1 g_1 + f_2 g_2 + f_3 g_3$$

Figure 8 shows multiplied components using equal colors.

Figure 8: Forming the dot product of two vectors.

Note the use of numeric subscripts, $f_1$, $f_2$, and $f_3$ rather than $f_x$, $f_y$, and $f_z$; it means the same thing. Numeric subscripts allow the three term sum above to be written more compactly as:

$$\vec f \cdot \vec g \equiv \sum_{\mbox{\scriptsize all }i} f_i g_i$$

The $\Sigma$ is called the “summation symbol.”

The dot (or “inner”) product of functions is defined in exactly the same way as for vectors, by multiplying values at the same $x$ position together and summing. But since there are infinitely many $x$-values, the sum becomes an integral:

$$f \cdot g = \int_{\mbox{\scriptsize all }x} f(x) g(x) {\rm d} x$$ (1)

as illustrated in figure 9.

Figure 9: Forming the inner product of two functions.

8.3 Application to the present problem

Now let us have a look at the separation of variables procedure for $u_{tt} = a^2 u_{xx}$. First of all, let us write this problem like $u_{tt}=Au$ where $A$ is the operator

$$A= a^2 \frac{\partial^2}{\partial x^2}$$

I want to convince you that the operator $A$ is really like a matrix. Indeed, a matrix A can transform any arbitrary vector $\vec f$ into a different vector $A\vec f$:

$$\vec f \quad \begin{picture}(100,0) \put(50,15){\makebox... ...0,2){\vector(1,0){100}} \end{picture} \quad \vec g = A \vec f$$

Similarly, the operator $A$ above transforms a function into another function, $a^2$ times its second order derivative:

$$f(x) \quad \begin{picture}(100,10) \put(50,15){\makebox(... ...tor(1,0){100}} \end{picture} \quad g(x) = A f(x) = a^2 f''(x)$$

Now I want to convince you that the operator $A$ above is not just a matrix, but a symmetric matrix. If a matrix $A$ is symmetric then for any two vectors $\vec f$ and $\vec g$,

$$\vec f^T (A\vec g) = (A\vec f)^T \vec g$$

or in other words, the dot product between $\vec f$ and $A\vec g$ is the same as that between $A\vec f$ and $\vec g$. For our operator $A$ we have:

$$f \cdot (A g) = \int_{\mbox{\scriptsize all }x} f(x) a^2 g''(x) {\rm d} x$$

and using two integrations by parts, (and the homogeneous boundary conditions,) this can be transformed into $Af \cdot g$. So $A$ must be symmetric.

Now it is no longer that surprising that we have only real eigenvalues; that is a general property of symmetric matrices.

And remember also that symmetric matrices have orthogonal eigenvectors, so the eigenfunctions $X_1$, $X_2$, $X_3$, ...of operator $A$ are going to be orthogonal. And a complete set, so we can write any function of $x$ in terms of these eigenfunctions, including the initial conditions $f$, $g$, and the solution $u$:

$$f=\sum f_n X_n \qquad g=\sum g_n X_n \qquad u=\sum u_n X_n$$

How do we get the $f_n$ given $f$? Well, we usually do not normalize the eigenfunctions to “length” one, so the unit function in the direction of an $X_n$ is:

$$e_n = \frac{1}{\sqrt{X_n\cdot X_n}}X_n$$

and the component of $f$ in the direction of this unit function, call it $c_n$, is found by taking a dot product:

$$c_n= \frac{X_n\cdot f}{\sqrt{X_n\cdot X_n}}$$

The component vector of $f$ in the direction of $X_n$ is then $c_n$ times the unit function:

$$\frac{X_n\cdot f}{\sqrt{X_n\cdot X_n}} e_n = \frac{X_n\cdot f}{X_n\cdot X_n} X_n$$

so the Fourier coefficient $f_n$ is

$$f_n = \frac{X_n\cdot f}{X_n\cdot X_n}$$

That is our old “orthogonality relation.”

Another way to understand the orthogonality relation is in terms of transformation matrices. Note that the $f_n$ are just the components of function $f$ when written in terms of the eigenfunctions. We get them by evaluating $P^{-1} f$, where the transformation matrix $P$ consists of the eigenfunctions as columns, and $P^{-1}$ is found as the transpose matrix (ignoring the lack of normalization of the eigenfunctions):

$$P=\left(X_1, X_2, X_3, \ldots\right) \qquad P^{-1}=\left(\begin{array}{c}X_1\ X_2\ X_3\ \vdots\end{array}\right)$$

It is seen that $P^{-1} f$ then works out to dot products between the $X_n$ and $f$.