# Teaching and motivating the use of Eigenvectors

I would like to know how to better demonstrate Eigenvectors. The texts that I have display the properties and methods to calculate them. There are plenty of great elementary examples to follow through before taking on larger matrices and applying them in software to large scale problems.

Regardless of this, there are many graduates working as analysts who still say that they know how to compute them but do not have a solid ground understanding of them.

How can they be better motivated?

• See the answer by Anchewski at matheducators.stackexchange.com/questions/520/…? – mweiss Aug 19 '15 at 15:36
• That is for eigenvalues, and captures a different aspect although closely related. – Vass Aug 19 '15 at 15:49
• Although the question is about eigenvalues, the answer I refer to is specifically about the eigenvectors (which is why I indicated to see the answer, rather than the question). – mweiss Aug 19 '15 at 15:49
• It is an interesting answer, and is enlightening, but I would like to know of a more elaborate and verbose treatment. My experience with teaching concise examples (however powerful and convincing they are) is that they miss question internalised in the students minds – Vass Aug 19 '15 at 15:59
• In that case I think you need to be much clearer about what you are asking about. What do you think the "question internalised in the students minds" is? What do you think is inadequate about specific motivating examples? – mweiss Aug 19 '15 at 16:00

I personally found Principal Component Analysis to be helpful in explaining the use of eigenvectors and eigenvalues. Jeff Jauregui at Union College wrote a great article on PCA that I've used in my Linear Algebra class. (The bird example is pure pedagogical gold.) You can find it here: www.math.union.edu/~jaureguj/PCA.pdf

Note: Our curriculum contains a decent amount of statistics early on, so my students breezed over the necessary stats knowledge.

Principal motivation behind eigenvectors is, of course, eigenspaces. And eigenspaces, in turn, are an important case of invariant subspaces and one of few constructs that we can make of a linear operator using pure linear algebra (without Euclidean norm, coordinates etc.). In some cases eigenspaces classify invariant subspaces. Suppose we don’t know anything about an abstract vector space. What is the only way to specify a linear operator? Multiplication by a fixed scalar, i.e. $$v\mapsto \lambda v\,,$$ of course. And finding subspaces where some not-so-simple operator has such form shows how to describe an operator using subspaces and scalars. When linear operator is diagonalizable (i.e. hasn’t Jordan blocks, that is a general position), it provides complete description of the operator, albeit over complex numbers (or another algebraically closed field).

Students should know that:

• An eigenvector multiplied by a non-zero scalar is an eigenvector again.
• A linear combination of eigenvectors with the same eigenvalue is an eigenvector again (if non-zero), with the same eigenvalue – that’s how eigenspaces arise.
• Linear span of some set of eigenvectors (or direct sum of eigenspaces) produces an invariant subspace, but not necessarily an eigenspace.

This is an archetypical situation that reproduces, to some extent, many times in algebra: decomposition to elementary structural blocks.

Some examples requested by the original poster. Let, over some F, $$A = \begin{pmatrix} 0 && 1 && 0 \\ 1 && 0 && 0 \\ 0 && 0 && 1 \end{pmatrix}.$$ The simplest choice of eigenvectors of $A$ is: $$\begin{pmatrix}1\\1\\0\end{pmatrix}, \begin{pmatrix}0\\0\\1\end{pmatrix}\ \text{for }\lambda=1; \quad \begin{pmatrix}1\\-1\\0\end{pmatrix}\ \text{for }\lambda=-1.$$ Then an arbitrary linear combination of two $\lambda=1$ eigenvectors, namely $$\begin{pmatrix}α\\α\\β\end{pmatrix},$$ describes the $\lambda=1$ eigenspace of $A$, a two-dimensional subspace of F3. Its non-zero elements are all $\lambda=1$ eigenvectors of $A$.
An arbitrary linear combination of the first and the third eigenvectors, namely $$\begin{pmatrix}α+γ\\α-γ\\0\end{pmatrix},$$ describes an $A$-invariant subspace, but not an eigenspace of $A$ (assuming 2 ≠ 0) because “1” and “−1” eigenvalues are mixed.

• deep and insightful. Can you expand on the second bullet point? Simple example please? – Vass Aug 24 '15 at 13:21
• @Vass: now with examples on second and third bullet points. – Incnis Mrsi Aug 24 '15 at 13:52

I find that the explanations in Treil's "Linear Algebra done Wrong" on all this are the clearest I've seen. But that could mean restructuring quite a bit of the rest of the course.