Consider a positive transition matrix A = [ a b c d ] , meaning that a, b, c, and d are positive numbers such that a + c = b + d = 1 . (The matrix in Exercise 24 has this form.) Verify that [ b c ] and [ 1 − 1 ] are eigenvectors of A . What are the associated eigenvalues? Is the absolute value of these eigenvalues more or less than 1? Sketch a phase portrait.
Consider a positive transition matrix A = [ a b c d ] , meaning that a, b, c, and d are positive numbers such that a + c = b + d = 1 . (The matrix in Exercise 24 has this form.) Verify that [ b c ] and [ 1 − 1 ] are eigenvectors of A . What are the associated eigenvalues? Is the absolute value of these eigenvalues more or less than 1? Sketch a phase portrait.
Solution Summary: The author explains that the vectors left[lb
Consider a positive transition matrix
A
=
[
a
b
c
d
]
, meaning that a, b, c, and d are positive numbers such that
a
+
c
=
b
+
d
=
1
. (The matrix in Exercise 24 has this form.) Verify that
[
b
c
]
and
[
1
−
1
]
are eigenvectors of A. What are the associated eigenvalues? Is the absolute value of these eigenvalues more or less than 1? Sketch a phase portrait.
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