2. How to locate the new axes

The key to finding the new basis is to compare what happens to vectors on and off the desired axes. In Figures and , what do you notice?

Figure: A diagonal transformation again

udir/diag2.eps

Figure: A diagonalizable transformation

udir/secret2.eps

Notice that a vector on either of the desired axes has an image along the same line. In other words, it has been multiplied by a scalar. In contrast, for a vector not on one of the desired axes, its image is not along the same line.

This is the key! For any matrix transformation $\tau _ A$, we look for vectors that get multiplied by a scalar. These tell us the new axes to use. As you will see, they are easy to compute.

Definition. A nonzero vector v with $\tau _ A ($v$) = \lambda$   v for some scalar $\lambda$ is called an eigenvector of $\tau _ A$. The scalar $\lambda$ is the corresponding eigenvalue¹.

Notes.

1. It is traditional to use the Greek letter $\lambda$ (lambda) for the scalar. This seems strange at first, but it's really helpful, in that when you see statements with $\lambda$ you immediately realize they are about eigenvalues.

2. Figures and show $T$ applied to just one vector not on the axes, but other vectors would behave similarly.

3. All we need is to choose one vector on each new axis for a new basis. Any nonzero vector along the axis will do.

4. The new axes are lines through the origin and so are subspaces. Therefore they are called eigenspaces for $\tau _ A$. We'll emphasize eigenvectors for the present, but it's actually the eigenspaces that are the neater concept.

5. In general, for a $2 \times 2$ matrix $A$, there can be either two, one, or no eigenspaces. If there are two, then you can make a new basis from two eigenvectors and get a diagonal matrix relative to the new basis. In this case, $A$ is said to be diagonalizable.

6. In the more general setting $T: V\rightarrow V$, where $V$ is a vector space, the concept is the same: If $T($v$) = \lambda$   v for a nonzero vector v, then v is an eigenvector and $\lambda$ is an eigenvalue.

7. For a matrix transformation $\tau _ A$, to say `` v is an eigenvector of $A$'' is the same thing as saying `` v is an eigenvector of $\tau _ A$.'' Obviously, v is an eigenvector of $A$ when v is nonzero and $A$   v$= \lambda$   v for some scalar $\lambda$.

8. An eigenvalue could be negative, so that $T($v$)$ points in the opposite direction from v. An eigenvalue can even be 0. (An eigenvector must be nonzero, though.)

9. There is still the question of how to compute eigenvectors and eigenvalues, even for $2 \times 2$ matrices; this question will be answered below.