Eigenvalues and Eigenvectors

In this section, we will define the eigenvalues and eigenvectors of a matrix, and see how to compute them. More theoretical properties will be taken up in the next section.

1. A=\begin{bmatrix} -5 & 8\\-4 & 7 \end{bmatrix}
2. For each eigenvalue of $A$, find the corresponding eigenspace.
3. For the polynomial $p(x)=3x^2-x+2$ and $A$ from above, compute $p(A)$.

We start with the principal definition for this chapter.

Before going any further, perhaps we should convince you that such things ever happen at all. Understand the next example, but do not concern yourself with where the pieces come from. We will have methods soon enough to be able to discover these eigenvectors ourselves.

Sage can compute eigenvalues and eigenvectors of matrices. We will see shortly that there are subtleties involved with using these routines, but here is a quick example to begin with. These two commands should be enough to get you started with most of the early examples in this section. See the end of the section for more comprehensive advice.

For a square matrix, the methods .eigenvalues() and .eigenvectors_right() will produce what you expect, though the format of the eigenvector output requires some explanation. Here is from the start of this chapter.

The three eigenvalues we know are included in the output of eigenvalues(), though for some reason the eigenvalue $\lambda=2$ shows up twice.

The output of the eigenvectors_right() method is a list of triples. Each triple begins with an eigenvalue. This is followed by a list of eigenvectors for that eigenvalue. Notice the first eigenvector is identical to the one we described in . The eigenvector for $\lambda=0$ is different, but is just a scalar multiple of the one from . For $\lambda=2$, we now get two eigenvectors, and neither looks like either of the ones from . (Hint: try writing the eigenvectors from the example as linear combinations of the two in the Sage output.) An explanation of the the third part of each triple (an integer) will have to wait, though it can be optionally surpressed if desired.

in Sage, so do not try to use this as a name for your eigenvalues.

hints at a number of intriguing properties, and there are many more. We will explore the general properties of eigenvalues and eigenvectors in , but in this section we will concern ourselves with the question of actually computing eigenvalues and eigenvectors. First we need a bit of background material on polynomials and matrices.

Condensed — the full section is in Beezer, A First Course in Linear Algebra.

A polynomial is a combination of powers, multiplication by scalar coefficients, and addition (with subtraction just being the inverse of addition). We never have occasion to divide when computing the value of a polynomial. So it is with matrices. We can add and subtract matrices, we can multiply matrices by scalars, and we can form powers of square matrices by repeated applications of matrix multiplication. We do not normally divide matrices (though sometimes we can multiply by an inverse). If a matrix is square, all the operations constituting a polynomial will preserve the size of the matrix. So it is natural to consider evaluating a polynomial with a matrix, effectively replacing the variable of the polynomial by a matrix. We will demonstrate with an example.

Condensed — the full section is in Beezer, A First Course in Linear Algebra.

Before we embark on computing eigenvalues and eigenvectors, we will prove that every matrix has at least one eigenvalue (and an eigenvector to go with it). Later, in , we will determine the maximum number of eigenvalues a matrix may have.

The determinant () will be a powerful tool in when it comes time to compute eigenvalues. However, it is possible, with some more advanced machinery, to compute eigenvalues without ever making use of the determinant. Sheldon Axler does just that in his book, Linear Algebra Done Right. Here and now, we give Axler's determinant-free proof that every matrix has an eigenvalue. The result is not too startling, but the proof is most enjoyable.

The proof of is constructive (it contains an unambiguous procedure that leads to an eigenvalue), but it is not meant to be practical. We will illustrate the theorem with an example, the purpose being to provide a companion for studying the proof and not to suggest this is the best procedure for computing an eigenvalue.

Condensed — the full section is in Beezer, A First Course in Linear Algebra.

Fortunately, we need not rely on the procedure of each time we need an eigenvalue. It is the determinant, and specifically , that provides the main tool for computing eigenvalues. Here is an informal sequence of equivalences that is the key to determining the eigenvalues and eigenvectors of a matrix, A\vect{x}=\lambda\vect{x}\iff A\vect{x}-\lambda I_n\vect{x}=\zerovector\iff \left(A-\lambda I_n\right)\vect{x}=\zerovector

So, for an eigenvalue $\lambda$ and associated eigenvector $\vect{x}\neq\zerovector$, the vector $\vect{x}$ will be a nonzero element of the null space of $A-\lambda I_n$, while the matrix $A-\lambda I_n$ will be singular and therefore have zero determinant. These ideas are made precise in and , but for now this brief discussion should suffice as motivation for the following definition and example.

The characteristic polynomial is our main computational tool for finding eigenvalues, and will sometimes be used to aid us in determining the properties of eigenvalues.

Let us now turn our attention to the computation of eigenvectors.

Condensed — the full section is in Beezer, A First Course in Linear Algebra.

There are no theorems in this section, just a selection of examples meant to illustrate the range of possibilities for the eigenvalues and eigenvectors of a matrix. These examples can all be done by hand, though the computation of the characteristic polynomial would be very time-consuming and error-prone. It can also be difficult to factor an arbitrary polynomial, though if we were to suggest that most of our eigenvalues are going to be integers, then it can be easier to hunt for roots. These examples are meant to look similar to a concatenation of , and . First, we will sneak in a pair of definitions so we can illustrate them throughout this sequence of examples.

Since an eigenvalue $\lambda$ is a root of the characteristic polynomial, there is always a factor of $(x-\lambda)$, and the algebraic multiplicity is just the power of this factor in a factorization of $\charpoly{A}{x}$. So in particular, $\algmult{A}{\lambda}\geq 1$. Compare the definition of algebraic multiplicity with the next definition.

Every eigenvalue must have at least one eigenvector, so the associated eigenspace cannot be trivial, and so $\geomult{A}{\lambda}\geq 1$.

Condensed — the full section is in Beezer, A First Course in Linear Algebra.