Eigenvalues and Eigenvectors
Consider the operator box illustrated in the figure 10-5 where the input to the
operator box is the n× 1 nonzero column vector x=col(x 1 , x 2 ,... , x n)and the output
from the operator box is the n× 1 column vector y=Ax were A is a n×nnonzero
constant matrix. The operator box is said to transform the nonzero column vector
xto the column vector yby matrix multiplication. For example, if n= 3 one would
have the situation illustrated.
Figure 10-5.
Transformation of vector xto vector y.
y 1
y 2
y 2
=
a 11 a 12 a 13
a 21 a 22 a 23
a 31 a 32 a 33
x 1
x 2
x 3
If there are special nonzero column vectors x such that the output y is pro-
portional to the input x, then these special vectors are called eigenvectors and the
proportionality constants are called eigenvalues. If the output yis proportional to
the nonzero input x, then the equation y=Ax =λx must be satisfied, where λis the
scalar proportionality constant. If the equation Ax =λx has nonzero solutions, then
one can write
Ax =λx =λ Ix
(A−λ I)x=[0]n× 1
a 11 −λ a 12... a 1 n
a 21 a 22 −λ... a 2 n..
.
..
.
... ..
.
an 1 an 2... a nn −λ
x 1
x 2..
.
xn
=
0
0..
.
0
(10 .29)Cramer’s^3 rule states that in order for this last equation to have a nonzero solution it
is required that the determinant of the unknowns x 1 , x 2 ,... , x nbe zero. This requires
that
det(A−λI ) =
∣∣
∣∣
∣∣
∣∣a 11 −λ a 12... a 13
a 21 a 22 −λ... a 2 n..
.
..
.
... ..
.
an 1 an 2... a nn −λ∣∣
∣∣
∣∣
∣∣= 0 (10 .30)(^3) Gabriel Cramer (1704-1752) A Swiss mathematician who studied determinants.