In all parts of this problem, consider an n × n matrix A such that all entries are positive and the sum of the entries in each row is 1 (meaning that A T is a positive transition matrix). a. Consider an eigenvector υ → of A with positive components. Show that the associated eigenvalue is less than or equal to 1. Hint : Consider the largest entry υ i of υ → . What can you say about the ith entry of A υ → ? b. Now we drop the requirement that the components of the eigenvector υ → be positive. Show that the associated eigenvalue is less than or equal to 1 in absolute value. c. Show that λ = − 1 fails to be an eigenvalue of A , and show that the eigenvectors with eigenvalue 1 are the vectors of the form [ c c ⋮ c ] Where c is nonzero.
In all parts of this problem, consider an n × n matrix A such that all entries are positive and the sum of the entries in each row is 1 (meaning that A T is a positive transition matrix). a. Consider an eigenvector υ → of A with positive components. Show that the associated eigenvalue is less than or equal to 1. Hint : Consider the largest entry υ i of υ → . What can you say about the ith entry of A υ → ? b. Now we drop the requirement that the components of the eigenvector υ → be positive. Show that the associated eigenvalue is less than or equal to 1 in absolute value. c. Show that λ = − 1 fails to be an eigenvalue of A , and show that the eigenvectors with eigenvalue 1 are the vectors of the form [ c c ⋮ c ] Where c is nonzero.
Solution Summary: The author explains that the associated eigenvalue is less than or equal to 1.
In all parts of this problem, consider an
n
×
n
matrix A such that all entries are positive and the sum of the entries in each row is 1 (meaning that
A
T
is a positive transition matrix). a. Consider an eigenvector
υ
→
of A with positive components. Show that the associated eigenvalue is less than or equal to 1. Hint: Consider the largest entry
υ
i
of
υ
→
. What can you say about the ith entry of
A
υ
→
? b. Now we drop the requirement that the components of the eigenvector
υ
→
be positive. Show that the associated eigenvalue is less than or equal to 1 in absolute value. c. Show that
λ
=
−
1
fails to be an eigenvalue of A, and show that the eigenvectors with eigenvalue 1 are the vectors of the form
[
c
c
⋮
c
]
Where c is nonzero.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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