21 votes
Accepted

What is a good motivation/showcase for a student for the study of eigenvalues?

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.
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18 votes

What is a good motivation/showcase for a student for the study of eigenvalues?

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 ...
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14 votes

What is a good motivation/showcase for a student for the study of eigenvalues?

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 ...
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12 votes

What is a good motivation/showcase for a student for the study of eigenvalues?

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 ...
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10 votes

What is a good motivation/showcase for a student for the study of eigenvalues?

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 ...
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9 votes
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What is the best way to intuitively explain what eigenvectors and eigenvalues are, AND their importance?

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 ...
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8 votes

What is a good motivation/showcase for a student for the study of eigenvalues?

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 $$ \...
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7 votes

What is a good motivation/showcase for a student for the study of eigenvalues?

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

What is a good motivation/showcase for a student for the study of eigenvalues?

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....
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6 votes

What is a good motivation/showcase for a student for the study of eigenvalues?

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^{\...
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6 votes

Computing eigenvalues by hand without determinants

I am going to just play here. Take $$ A = \begin{bmatrix} 1 & 2 \\ 3 & -4 \end{bmatrix} $$ The "Linear Algebra Done Right" idea for establishing existence of eigenvectors is that an ...
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5 votes

What is the best way to intuitively explain what eigenvectors and eigenvalues are, AND their importance?

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 ...
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5 votes

What is the best way to intuitively explain what eigenvectors and eigenvalues are, AND their importance?

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 ...
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4 votes

Computing eigenvalues by hand without determinants

Students tend to remember what they do much better than what they're told. If this is a student's first introduction to linear algebra, what's more important for them to remember about eigenvalues? I'...
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4 votes

What is the best way to intuitively explain what eigenvectors and eigenvalues are, AND their importance?

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 ...
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4 votes

What is a good motivation/showcase for a student for the study of eigenvalues?

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 ...
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4 votes

What is a good motivation/showcase for a student for the study of eigenvalues?

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 ...
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3 votes

What is the best way to intuitively explain what eigenvectors and eigenvalues are, AND their importance?

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 ...
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3 votes

Computing eigenvalues by hand without determinants

Presumably, on a homework or quiz, the matrices involved with be 2x2. Let's look at the bookkeeping labor involved for this example: $$A=\pmatrix{\ \ 3&1\\ -4&1}$$ We need to row reduce $$\...
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2 votes

What is a good motivation/showcase for a student for the study of eigenvalues?

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 ...
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2 votes

What is the best way to intuitively explain what eigenvectors and eigenvalues are, AND their importance?

As a video called "The 4 Ways to Tell if a Matrix is Diagonalizable [Passing Linear Algebra]" by the YouTube channel "STEM Support" mentions, Eigenvalues and Eigenvectors are ...
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1 vote

What is the best way to intuitively explain what eigenvectors and eigenvalues are, AND their importance?

Many other answers mentioned ODE, but the discrete version seems simpler to introduce at an early stage. One could use system of sequences to show a magic trick and then explain it with eigenvectors, ...
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1 vote

What is the best way to intuitively explain what eigenvectors and eigenvalues are, AND their importance?

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 ...
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1 vote

What is the best way to intuitively explain what eigenvectors and eigenvalues are, AND their importance?

For students of all ages, I would go the “dynamical system road” with pictures (never tried it though, mostly by lack of opportunity). Introduce a problem that is in fact a simple recursive vector-...
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1 vote

What is the best way to intuitively explain what eigenvectors and eigenvalues are, AND their importance?

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" (...
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