
# 12.1: Invariant Directions

Have a look at the linear transformation $$L$$ depicted below:

It was picked at random by choosing a pair of vectors $$L(e_{1})$$ and $$L(e_{2})$$ as the outputs of $$L$$ acting on the canonical basis vectors.  Notice how the unit square with a corner at the origin is mapped to a parallelogram.  The second line of the picture shows these superimposed on one another. Now look at the second picture on that line. There, two vectors $$f_{1}$$ and $$f_{2}$$ have been carefully chosen such that if the inputs into $$L$$ are in the parallelogram spanned by $$f_{1}$$ and $$f_{2}$$, the outputs also form a parallelogram with edges lying along the same two directions. Clearly this is a very special situation that should correspond to  interesting properties of $$L$$.

Now lets try an explicit example to see if we can achieve the last picture:

Example 116

Consider the linear transformation $$L$$ such that $$L\begin{pmatrix}1\\0\end{pmatrix}=\begin{pmatrix}-4\\-10\end{pmatrix}\, \mbox{ and }\, L\begin{pmatrix}0\\1\end{pmatrix}=\begin{pmatrix}3\\7\end{pmatrix}\, ,$$ so that the matrix of $$L$$ is

$$\begin{pmatrix} -4 & 3 \\ -10 & 7 \\ \end{pmatrix}\, .$$

Recall that a vector is a direction and a magnitude; $$L$$ applied to $$\begin{pmatrix}1\\0\end{pmatrix}$$ or $$\begin{pmatrix}0\\1\end{pmatrix}$$ changes both the direction and the magnitude of the vectors given to it.

Notice that $$L\begin{pmatrix}3\\5\end{pmatrix}=\begin{pmatrix}-4\cdot 3+3\cdot 5 \\ -10\cdot 3+7\cdot 5\end{pmatrix}=\begin{pmatrix}3\\5\end{pmatrix}\, .$$  Then $$L$$ fixes the  direction (and actually also the magnitude) of the vector $$v_{1}=\begin{pmatrix}3\\5\end{pmatrix}$$.

Now, notice that any vector with the same direction as $$v_{1}$$ can be written as $$cv_{1}$$ for some constant $$c$$.  Then $$L(cv_{1})=cL(v_{1})=cv_{1}$$, so $$L$$ fixes every vector pointing in the same direction as $$v_{1}$$.

Also notice that $$L\begin{pmatrix}1\\2\end{pmatrix}=\begin{pmatrix}-4\cdot 1+3\cdot 2 \\ -10\cdot 1+7\cdot 2\end{pmatrix}=\begin{pmatrix}2\\4\end{pmatrix}=2\begin{pmatrix}1\\2\end{pmatrix}\, ,$$ so $$L$$ fixes the direction of the vector $$v_{2}=\begin{pmatrix}1\\2\end{pmatrix}$$ but stretches $$v_{2}$$ by a factor of $$2$$.  Now notice that for any constant $$c$$, $$L(cv_{2})=cL(v_{2})=2cv_{2}$$.  Then $$L$$ stretches every vector pointing in the same direction as $$v_{2}$$ by a factor of $$2$$.

In short, given a linear transformation $$L$$ it is sometimes possible to find a vector $$v\neq 0$$ and constant $$\lambda\neq 0$$ such that $$Lv=\lambda v.$$

We call the direction of the vector $$v$$ an $$\textit{invariant direction}$$.  In fact, any vector pointing in the same direction also satisfies this equation because $$L(cv)=cL(v)=\lambda cv$$.  More generally, any $$\textit{non-zero}$$ vector $$v$$ that solves

$Lv=\lambda v$

is called an $$\textit{eigenvector}$$ of $$L$$, and $$\lambda$$ (which now need not be zero) is an $$\textit{eigenvalue}$$.  Since the direction is all we really care about here, then any other vector $$cv$$ (so long as $$c\neq 0$$) is an equally good choice of eigenvector. Notice that the relation "$$u$$ and $$v$$ point in the same direction'' is an equivalence relation.

In our example of the linear transformation $$L$$ with matrix

$$\begin{pmatrix} -4 & 3 \\ -10 & 7 \\ \end{pmatrix}\, ,$$

we have seen that $$L$$ enjoys the property of having two invariant directions, represented by eigenvectors $$v_{1}$$ and $$v_{2}$$ with eigenvalues $$1$$ and $$2$$, respectively.

It would be very convenient if we could write any vector $$w$$ as a linear combination of $$v_{1}$$ and $$v_{2}$$.  Suppose $$w=rv_{1}+sv_{2}$$ for some constants $$r$$ and $$s$$.  Then:

$L(w)=L(rv_{1}+sv_{2})=rL(v_{1})+sL(v_{2})=rv_{1}+2sv_{2}.$

Now $$L$$ just multiplies the number $$r$$ by $$1$$ and the number $$s$$ by $$2$$.  If we could write this as a matrix, it would look like:

$\begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}\begin{pmatrix}s\\t\end{pmatrix}$
which is much slicker than the usual scenario  $$L\!\begin{pmatrix} x\\ y \end{pmatrix}\!=\!\begin{pmatrix} \!a&b\! \\ \!c&d\! \end{pmatrix} \! \!\begin{pmatrix} x \\ y \end{pmatrix}\!=\! \begin{pmatrix} \!ax+by\! \\ \!cx+dy\! \end{pmatrix}\, .$$

Here, $$s$$ and $$t$$ give the coordinates of $$w$$ in terms of the vectors $$v_{1}$$ and $$v_{2}$$.  In the previous example, we multiplied the vector by the matrix $$L$$ and came up with a complicated expression.  In these coordinates, we see that $$L$$ has a very simple $$\textit{diagonal matrix}$$, whose diagonal entries are exactly the $$\textit{eigenvalues}$$ of $$L$$.

This process is called $$\textit{diagonalization}$$. It makes complicated linear systems much easier to analyze.

Now that we've seen what eigenvalues and eigenvectors are, there are a number of questions that need to be answered.

1. 1.    How do we find eigenvectors and their eigenvalues?
2. 2.    How many eigenvalues and (independent) eigenvectors does a given linear transformation have?
3. 3.    When can a linear transformation be diagonalized?

We'll start by trying to find the eigenvectors for a linear transformation.

Example 117

Let $$L \colon \Re^{2}\rightarrow \Re^{2}$$ such that $$L(x,y)=(2x+2y, 16x+6y)$$.  First, we  find the matrix of $$L$$:

$\begin{pmatrix}x\\y\end{pmatrix}\stackrel{L}{\longmapsto} \begin{pmatrix} 2 & 2 \\ 16 & 6 \end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}.$

We want to find an invariant direction $$v=\begin{pmatrix}x\\y\end{pmatrix}$$ such that

$Lv=\lambda v$

or, in matrix notation,
\begin{equation*}
\begin{array}{lrcl}
&\begin{pmatrix}
2 & 2 \\
16 & 6
\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}&=&\lambda \begin{pmatrix}x\\y\end{pmatrix} \\
\Leftrightarrow &\begin{pmatrix}
2 & 2 \\
16 & 6
\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}&=&\begin{pmatrix}
\lambda & 0 \\
0 & \lambda
\end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix} \\
\Leftrightarrow& \begin{pmatrix}
2-\lambda & 2 \\
16 & 6-\lambda
\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}&=& \begin{pmatrix}0\\0\end{pmatrix}\, .
\end{array}
\end{equation*}
This is a homogeneous system, so it only has solutions when the matrix $$\begin{pmatrix} 2-\lambda & 2 \\ 16 & 6-\lambda \end{pmatrix}$$ is singular.  In other words,
\begin{equation*}
\begin{array}{lrcl}
&\det \begin{pmatrix}
2-\lambda & 2 \\
16 & 6-\lambda
\end{pmatrix}&=&0 \\
\Leftrightarrow& (2-\lambda)(6-\lambda)-32&=&0 \\
\Leftrightarrow &\lambda^{2}-8\lambda-20&=&0\\
\Leftrightarrow &(\lambda-10)(\lambda+2)&=&0
\end{array}
\end{equation*}

For any square $$n\times n$$ matrix $$M$$, the polynomial in $$\lambda$$ given by $$P_{M}(\lambda)=\det (\lambda I-M)=(-1)^{n} \det (M-\lambda I)$$ is called the $$\textit{characteristic polynomial}$$ of $$M$$, and its roots are the eigenvalues of $$M$$.

In this case, we see that $$L$$ has two eigenvalues, $$\lambda_{1}=10$$ and $$\lambda_{2}=-2$$.  To find the eigenvectors, we need to deal with these two cases separately.  To do so, we solve the linear system $$\begin{pmatrix} 2-\lambda & 2 \\ 16 & 6-\lambda \end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}= \begin{pmatrix}0\\0\end{pmatrix}$$ with the particular eigenvalue $$\lambda$$ plugged in to the matrix.

1.    [$$\underline{\lambda=10}$$:]  We solve the linear system

$\begin{pmatrix} -8 & 2 \\ 16 & -4 \end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}= \begin{pmatrix}0\\0\end{pmatrix}.$
Both equations say that $$y=4x$$, so any vector $$\begin{pmatrix}x\\4x\end{pmatrix}$$ will do.  Since we only need the direction of the eigenvector, we can pick a value for $$x$$.  Setting $$x=1$$ is convenient, and gives the eigenvector $$v_{1}=\begin{pmatrix}1\\4\end{pmatrix}$$.

2.    [$$\underline{\lambda=-2}$$:]  We solve the linear system

$\begin{pmatrix} 4 & 2 \\ 16 & 8 \end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}= \begin{pmatrix}0\\0\end{pmatrix}.$
Here again both equations agree, because we chose $$\lambda$$ to make the system singular.  We see that $$y=-2x$$ works, so we can choose $$v_{2}=\begin{pmatrix}1\\-2\end{pmatrix}$$.

Our process was the following:

1. Find the characteristic polynomial of the matrix $$M$$ for $$L$$, given by $$\det (\lambda I-M)$$.
2. Find the roots of the characteristic polynomial; these are the eigenvalues of $$L$$.
3. For each eigenvalue $$\lambda_{i}$$, solve the linear system $$(M-\lambda_{i} I)v=0$$ to obtain an eigenvector $$v$$ associated to $$\lambda_{i}$$.