Page 230 - 35Linear Algebra

230 Eigenvalues and Eigenvectors

direction also satisfies this equation because L(cv) = cL(v) = λcv. More
generally, any non-zero vector v that solves

L(v) = λv

is called an eigenvector of L, and λ (which now need not be zero) is an
eigenvalue. Since the direction is all we really care about here, then any other
vector cv (so long as c 6= 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

−4 3
,
−10 7
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:

1 0 s
0 2 t

which is much slicker than the usual scenario

x a b x ax + by
L = = .
y c d y cx + dy
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 diagonal matrix, whose diagonal entries are exactly the eigenvalues
of L.
This process is called diagonalization. It makes complicated linear sys-
tems much easier to analyze.

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