Applications of spectral theorem My question is quite simple, I'm looking for easy applications of the spectral theorem, i.e., hermitian matrices are diagonalizable to show to my students of linear algebra.
I've already found some applications of the spectral theorem, but not simple applications.
I'm looking for also simple applications to related fields such as economics, etc...
Thanks in advance
 A: -Arguably the most important application of the spectral theorem is the second derivative test in multivariable calculus.  The Taylor series of a function $f(x,y)$ is:
$$f(x,y) = f(a,b) + \nabla f(a,b) \cdot (x-a, y-b) + Qf(a,b)(x-a, x-b) + \ldots$$
where $Q$ is the quadratic form associated to the Hessian of $f$.  The spectral theorem says that $Q$ can be diagonalized, and this is how one decides if $f$ has a local min/max at a critical point.
-Covariance matrices: if $X$ is a vector valued random variable with identically distributed - but not necessarily independent - components then $Cov(X_i, X_j)$ is symmetric, and the fact that it can be diagonalized says that there is a change of coordinates which makes the components of $X$ uncorrelated. This is particularly useful for normal random variables where uncorrelated implies independent - the conclusion is that a vector valued random variable whose components are Gaussians can be transformed into a random variable whose components are i.i.d. Gaussians.
-Principle axis theorem: assume a rigid body moves through space, and a reference frame is chosen so that the center of mass remains fixed.  Then the motion of the body is determined by the moment of inertia tensor which is an anti-symmetric matrix.  The fact that it can be orthogonally diagonalized says that there are three perpendicular axis around which the rigid body is rotating.
A: An important application is in Morse theory, and in the analysis of critical points more generally: the Hessian of a smooth function is symmetric, and so can be diagonalized by an orthogonal matrix. The number of negative eigenvalues of the Hessian at a non-degenerate critical point is called the index of the critical point, which has important topological consequences when the function is defined on a smooth manifold. By "the analysis of critical points more generally," I just mean the second derivative test, and quadratic approximations.
I'm assuming you're also aware of the importance of the spectral theorem in the analysis of quadratic forms, and therefore of its importance in statistics and econometrics. See this for example, where Michael Hardy discusses "spectral decompositions in statistics."
The coolest application, to my mind, is in quantum mechanics, where observables are associated with Hermitian operators, the eigenvalues (more generally, the "spectrum") of which give the possible values of the observable. The spectral theorem ensures the eigenvalues are real, and hence "observable."
A: In quantum mechanics (von-Neumann style), observables=adjoint operators are independent if they commute=are jointly diagonalizable. Heisenberg uncertainity principle says that not all observables can be jointly diagonalized. Hence your system must be non-commutative.
A: Another application is that, you can easily compute $f(A)$ for any polynomial $f$. That is what is the final motivation for the spectral theorem in infinite dimensions.
