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Obtaining formulas for the $n$-th term in a linear recurrence, such as Fibonacci numbers, is one application that certainly does not overtly mention linear algebra in the set-up.

18

I like Markov chains and Google PageRank (which is essentially a special kind of Markov chain). It doesn't take very long to explain and motivate Markov chains and to argue that the probability distribution at time $n$ is the $n$'th power of the transition matrix times the distribution at time $0$. You can then start talking about how to calculate powers ...

14

Imagine a linear mapping $f: R^2 \to R^2, e_1 \mapsto (1.5, 0.5), e_2 \mapsto (0.5, 1.5)$. (As long as $R$ contains the numbers $1.5$ and $0.5$, it could be any ring. The real numbers serve as the most convenient example, however.) Can we "see", what the mapping does? Can we "see" what $f^5$ does? Given a basis of the two eigenvectors, $(1,-1), (1,1)$ we ...

12

I have always found the standard motivations for eigenvalues to be a little artificial. The primary application for eigenvalues is ultimately diagonalization and there are several ways you could try to motivate diagonalization: Taking large powers of matrices seems to be a popular one. But its not immediately obvious what this is used for. An extension of ...

10

For a real showcase, I recommend a scenario where resonance frequencies play a role. Suspension bridges are real-world objects which are delicate enough that soldiers are usually not allowed to march over bridges (the German traffic law StVO states this in § 27 Abs. 6). The reason for this is that small, regular excitations with a certain frequency will ...

8

One of the standard examples is from stability analysis of dynamical systems. In the linear approximation you ask whether the $0$ solution is stable for $$X_{j} = A X_{j-1}$$ (discrete time) or $$\frac{d}{dt} X = B X$$ (continuous time).

6

Suppose for simplicity that the matrix $A$ of some linear mapping $\mathbb{R}^n \to \mathbb{R}^n$ is symmetric (hence diagonalizable) and all the eigenvalues $\lambda_1,\ldots,\lambda_n$ are different. Then, there is an orthogonal base such that in that base $$\text{A behaves like scaling with factors \lambda_1,\ldots,\lambda_n respectively.}$$ In ...

6

One clear motivation is the relationship to optimization: The largest eigenvalue of a real symmetric matrix is the maximum value of the (normalized) quadratic form $$\frac{x^{\mathsf T} A x}{x^{\mathsf T} x}$$ This is used for instance in spectral graph theory and elsewhere.

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Another cool application is to figuring out whether a given matrix is positive definite. This is useful for the second derivative test, for example.

6

Here's an example I use for myself. I don't teach this topic in regular class but I have used this example in private conversations with advanced students. Think of an object (perhaps a globe) that is stretched on one or more directions then rotated in various ways and perhaps reflected. We can show that at least one line through the object is either still ...

5

I think a good motivation is the idea of dynamical systems and stability a la Markov chains. If we have a system which can be modelled by taking a vector of data $v(0)$ and then some matrix $A$ we have $v(t) = Av(t-1)$. Observe that such a system is in some sense stable, and will undergo consistent exponential growth if it is a eigenvector. Even more, one ...

5

Although the following quantum-ish-physics-y "explanation" begs-the-question in several ways, it is genuine, and may convey something to students: given a linear operator (a.k.a. "matrix"), an eigenvector is "a pure state" (of what, we don't quite ask), meaning that the operator acts on it in an especially simple fashion. In good situations, a general "state"...

4

Another way: You can introduce students to the recurrence relation: $$a_{n + 1} = a_n + 2b_n$$ $$b_{n + 1} = a_n + b_n$$ And show that this recurrent relation can be represented as a matrix multiplication: $$v_{n+1} = Av_n$$ where $A = \binom{1 \ \ 2}{1 \ \ 1}$ and $v_n = \binom{a_n}{b_n}$. Then show (just demonstrate, not prove) that $\lim_{n \... 4 Markov chains have already been mentioned, that is a nice direction. Of course, zero is an eigenvalue iff$A$is singular. I'm not sure I'm creative enough to take this as a start point for motivating the eigenvector concept. Another way to discover them is simply by taking a point$x_o$and a matrix$A$and calculating$x_1=Ax_o$,...$x_k = Ax_{k-1}$. ... 4 To be perfectly honest, the applications of eigenvalues tends to be much more complex than the actual theory of finding eigenvalues itself. The first time I found a practical application of them was in Differential Equations, particularly solving the problems of $$y' = Ax$$ whereas$A$is a constant matrix$y$is a vector function of$x$which itself is a ... 2 This may be too advanced for some students in a first-semester linear algebra course, but those who have had some Physics may be impressed by the diagonalization of a moment of inertia tensor. Take some irregularly-shaped 3-dimensional object, and form a$3 \times 3$matrix describing all of the components of the moments of inertia around various axes; the ... 1 Familiar, everyday applications of eigenpairs abound: for young musicians: the natural frequencies of a musical instrument string are eigenvalues (actually they're the squares, but, for teaching purposes...). The vibration modes are the eigenvectors. The lowest eigenvalue for the string is the fundamental frequency, the higher values are harmonics. ... 1 As illustrated by other answers, Markov processes and linear ODE systems are the foremost applications. I like to mess a little with students explaining that the "only" thing that Google machines do all day is compute one large eigenvector over and over agin. But to get a feeling for why are eigenvectors useful, think simply of base changes: Let's say there ... 1 Eigenvectors are vectors that map onto themselves. (Eigen= "oneself" in German.) To accomplish this, they are defined by vectors of eigenvalues that "solve" the so-called "characteristic equation" (which defines them) for the value zero. This equation is derived from:$A v = \lambda v$, where A is the "transformation matrix, v is the eigenvector, and$\...

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