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In-Depth Information
Let
M
be the matrix whose eigenpairs we would like to find. Start with any nonzero vec-
tor
x
0
and then iterate:
where
∥
N
∥
for a matrix or vector
N
denotes the
Frobenius norm
; that is, the square root
of the sum of the squares of the elements of
N
. We multiply the current vector
x
k
by the
matrix
M
until convergence (i.e.,
∥
x
k
−
x
k
+1
∥
is less than some small, chosen constant).
Let
x
be
x
k
for that value of
k
at which convergence is obtained. Then
x
is (approximately)
the principal eigenvector of
M
. To obtain the corresponding eigenvalue we simply compute
λ
1
=
x
T
M
x
, which is the equation
M
x
= λ
x
solved for λ, since
x
is a unit vector.
EXAMPLE
11.3 Take the matrix from
Example 11.2
:
and let us start with
x
0
a vector with 1 for both components. To compute
x
1
, we multiply
M
x
0
to get
The Frobenius norm of the result is
We obtain
x
1
by dividing 5 and 8 by 9.434;
that is:
For the next iteration, we compute
The Frobenius norm of the result is 6.971, so we divide to obtain
We are converging toward a normal vector whose second component is twice the first. That
is, the limiting value of the vector that we obtain by power iteration is the principal eigen-
vector:
Finally, we compute the principal eigenvalue by
