Fundamentals of Matrix Algebra, 2011a

Chapter 4
Eigenvalues and Eigenvectors
We start by considering the matrix A and vector ⃗x as given below. 1
[ ] [ ]
1 4
1
A =
⃗x =
2 3
1
Mulplying A⃗x gives:
[ ] [ ] 1 4 1
A⃗x =
2 3 1
[ ] 5
=
5
[ ] 1
= 5 !
1
Wow! It looks like mulplying A⃗x is the same as 5⃗x! This makes us wonder lots
of things: Is this the only case in the world where something like this happens? 2 Is A
somehow a special matrix, and A⃗x = 5⃗x for any vector ⃗x we pick? 3 Or maybe ⃗x was a
special vector, and no maer what 2×2 matrix A we picked, we would have A⃗x = 5⃗x. 4
A more likely explanaon is this: given the matrix A, the number 5 and the vector
⃗x formed a special pair that happened to work together in a nice way. It is then natural
to wonder if other “special” pairs exist. For instance, could we find a vector ⃗x where
A⃗x = 3⃗x?
This equaon is hard to solve at first; we are not used to matrix equaons where
⃗x appears on both sides of “=.” Therefore we put off solving this for just a moment to
state a definion and make a few comments.
.
Ḋefinion 27
Eigenvalues and Eigenvectors
.
Let A be an n × n matrix, ⃗x a nonzero n × 1 column vector
and λ a scalar. If
A⃗x = λ⃗x,
then ⃗x is an eigenvector of A and λ is an eigenvalue of A.
The word “eigen” is German for “proper” or “characterisc.” Therefore, an eigenvector
of A is a “characterisc vector of A.” This vector tells us something about A.
Why do we use the Greek leer λ (lambda)? It is pure tradion. Above, we used a
to represent the unknown scalar, since we are used to that notaon. We now switch
to λ because that is how everyone else does it. 5 Don’t get hung up on this; λ is just a
number.
1 Recall this matrix and vector were used in Example 40 on page 75.
2 Probably not.
3 Probably not.
4 See footnote 2.
5 An example of mathemacal peer pressure.
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