Page 229 - 35Linear Algebra

12.1 Invariant Directions 229

Figure 12.1: The eigenvalue–eigenvector equation is probably the most im-
portant one in linear algebra.

3
Then L fixes the direction (and actually also the magnitude) of the vector v 1 = .
5

Reading homework: problem 1

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

1 −4 · 1 + 3 · 2 2 1
L = = = 2 ,
2 −10 · 1 + 7 · 2 4 2

1
so L fixes the direction of the vector v 2 = but stretches v 2 by a factor of 2.
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 6= 0 and constant λ 6= 0 such that Lv = λv. We call the direction of
the vector v an invariant direction. In fact, any vector pointing in the same

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