TITLE: What are all the generalizations needed to pass from finite dimensional linear algebra with matrices to fourier series and pdes?
QUESTION [35 upvotes]: I've studied Linear Algebra on finite dimensions and now I'm studying fourier series, sturm-liouville problems, pdes etc. However none of our lecturers made any connection between linear algebra an this. I think this is a very big mistake because I see many questions here where everyone just talks about these topics as generalizations of simple linear algebra.
Hence what are the generalizations? For example I think a matrix becomes a function in infinite dimension, but that's about it.

REPLY [30 votes]: Actually, separation of variables and orthogonal function expansions in the eigenfunctions of a Sturm-Liouville problem came before finite-dimensional linear algebra and the study of selfadjoint matrices. So, these topics are not generalizations of the linear algebra topics--they came first, which is one of the quirks of the subject. Again, infinite-dimensional orthogonal function expansions predated finite-dimensional linear algebra and the study of symmetric and Hermitian matrices on finite-dimensional Euclidean space. The infinite-dimensional studies drove the finite-dimensional.
If you have a selfadjoint square matrix $A$ on an $N$ dimensional real or complex space, then there is a finite number of eigenvalues $\lambda$ for which $A-\lambda I$ has a non-trivial null space. These eigenvalues $\{ \lambda_1,\lambda_2,\cdots,\lambda_n\}$ are the roots of the characteristic polynomial $p(\lambda)=\mbox{det}(A-\lambda I)$. For any such $\lambda_k$, a non-trivial solutions of $(A-\lambda_k I)x=0$ is an eigenvector with eigenvalue $\lambda_k$. Automatically (and this came from Sturm-Liouville theory,) the eigenvalues are real and the eigenvectors associated with different eigenvalues are automatically orthogonal to each other. For example, using the complex inner product, if $Ax=\lambda x$ for some $\lambda\in\mathbb{C}$ and $x \ne 0$, then $\lambda$ is real because the symmetry of $A$ gives
$$
     (\lambda-\overline{\lambda})(x,x)=(Ax,x)-(x,Ax) = 0 \implies \lambda=\overline{\lambda}.
$$
And, if $Ax_1 = \lambda_1 x_1$, $Ax_2=\lambda_2 x_2$ with $\lambda_1 \ne \lambda_2$, then $(x_1,x_2)=0$ because the symmetry of $A$ gives
$$
      (\lambda_1-\lambda_2)(x_1,x_2)=(Ax_1,x_2)-(x_1,Ax_2) = 0.
$$
Now, if you perform Gram-Schmidt on the different eigenspaces $\mbox{ker}(A-\lambda_k I)$, then you obtain an orthonormal basis of eigenvectors of $A$, which allows you to expand
$$
          x = \sum_{k=1}^{n} \sum_{j=1}^{n_k}\langle x,e_{k,j}\rangle e_{k,j}
$$
and to then represent $A$ in a simple diagonal form:
$$
          A x = \sum_{k=1}^{n} \sum_{j=1}^{n_k}\lambda_k \langle x,e_{k,j}\rangle e_{k,j}
$$
All of this analysis grew out of Fourier Analysis. Fourier Analysis is not a generalization of such, but this analysis is the specialization of Fourier Analysis ideas to a finite-dimensional setting.
Fourier Analysis on $[-\pi,\pi]$ may be viewed as the eigenfunction analysis of differentiation operator $A = -\frac{d^2}{dx^2}$ on the domain of functions $f$ with two derivatives in $L^2$ and satisfying $f(-\pi)=f(\pi)$, $f'(-\pi)=f'(\pi)$. The operator $A$ is symmetric with respect the integral inner product
$$
             (f,g) = \int_{-\pi}^{\pi}f(t)\overline{g(t)}dt.
$$
That is, $(Af,g)=(f,Ag)$ for all $f,g\in\mathcal{D}(A)$. Because of this, the eigenvalues of $A$ must be real (same argument as above) and the eigenfunctions corresponding to different eigenvalues must be orthogonal. The eigenfunctions are
$$
             1,\cos(nx),\sin(nx),\;\;\; n=1,2,3,\cdots.
$$
The eigenvalues are $0,1^2,2^2,3^2,\cdots,n^2,\cdots$. For example,
$$
                  L\cos(nx) = n^2 \cos(nx),\;\;\; L\sin(nx) = n^2 \sin(nx).
$$
The eigenspace for $n=0$ is one-dimensional with $L 1 = 0 \cdot 1$ (function $1$ and eigenvalue $0$.) The eigenspace for $n\ne 0$ is two-dimension and is spanned by $\sin(nx),\cos(nx)$. It is automatic that $\sin(nx),\cos(nx)$ are mutually orthogonal to $\sin(mx),\cos(mx)$ whenever $n\ne m$. And $\sin(nx),\cos(nx)$ are mutually orthogonal because of how they were chosen (nothing automatic.) The normalized eigenfunctions are
\begin{align}
        e_0 & = \frac{1}{\sqrt{2\pi}}, \\
        e_{1,1} & = \frac{1}{\sqrt{\pi}}\cos(x), e_{1,2}=\frac{1}{\sqrt{\pi}}\sin(x), \\
   e_{2,1} & =\frac{1}{\sqrt{\pi}}\cos(2x),e_{2,2}=\frac{1}{\sqrt{\pi}}\sin(2x),\;\;\; \\
   \vdots & = \vdots
\end{align}
The orthogonal function expansion of $f\in L^2[-\pi,\pi]$ in these eigenfunctions of $A$ is "the Fourier series" of $f$. This expansion always converges to $f$ in the norm of $L^2[-\pi,\pi]$. This constitutes an orthogonal diagonalization of the symmetric operator $A=-\frac{d^2}{dx^2}$, similar to the matrix case.
Other Sturm-Liouville problems exist for $\frac{d^2}{dx^2}$. For example,
$$
                      Lf = -\frac{d^2}{dx^2}f \\ 
                \cos\alpha f(a)+\sin\alpha f'(a) = 0 \\
                \cos\beta f(b) + \sin\beta f'(b) = 0
$$
The domain for $L$ here includes the above separated endpoint conditions. Again there is a discrete set of eigenvalues and all of the eigenspaces are one-dimensional. The eigenfunctions corresponding to these eigenvalues are all one-dimensional, and every $f\in L^2[a,b]$ can be expanded in a "Fourier series" of these orthogonal eigenfunctions. $L$ is symmetric again; so the eignevalues are real and the eigenfunctions corresponding to distinct eigenvalues are mutually orthogonal. These expansions include sine expansions, cosine expansions, and a wealth of other trigonometric function expansions where the eigenvalues are not at all evenly spaced. And all of the resulting Fourier expansions converge in $L^2$ to the original function.
The generalizations needed: Linear space, inner product and orthogonality, norm, approximation in norm and topology, symmetric and selfadjoint operators, orthogonal eigenvector expansions, differential operators, closed densely-defined unbounded operators, spectrum.
I've added a chart of development from Dieudonne's "A History of Functional Analysis" so that you can see how the infinite-dimensional gave rise to the finite-dimensional. This is very upside down, compared to most Math development. The most abstract came first and filtered down for a long before reversing course.