Is There A Steady State? So far, we've seen that for a conservative state transition matrix A, we can find the eigenvector, v, corre- sponding to the eigenvalue 2 = 1. This vector is the steady state since Av=v. However, we''ve so far taken for granted that the state transition matrix even has the eigenvalue 2 = 1. Let's try to prove this fact. (a) Show that if 2 is an eigenvalue of a matrix A, then it is also an eigenvalue of the matrix A". Hint: The determinants of A and A' are the same. This is because the volumes which these matrices represent are the same. (b) Let a square matrix A have, for each row, entries that sum to one. Show that 1 = [1 1 .. an eigenvector of A. What is the corresponding eigenvalue? 1]" is . (c) Let's put it together now. From the previous two parts, show that any conservative state transition matrix will have the eigenvalue 1 = 1. Recall that conservative state transition matrices have, for each column, entries that sum to 1.
Is There A Steady State? So far, we've seen that for a conservative state transition matrix A, we can find the eigenvector, v, corre- sponding to the eigenvalue 2 = 1. This vector is the steady state since Av=v. However, we''ve so far taken for granted that the state transition matrix even has the eigenvalue 2 = 1. Let's try to prove this fact. (a) Show that if 2 is an eigenvalue of a matrix A, then it is also an eigenvalue of the matrix A". Hint: The determinants of A and A' are the same. This is because the volumes which these matrices represent are the same. (b) Let a square matrix A have, for each row, entries that sum to one. Show that 1 = [1 1 .. an eigenvector of A. What is the corresponding eigenvalue? 1]" is . (c) Let's put it together now. From the previous two parts, show that any conservative state transition matrix will have the eigenvalue 1 = 1. Recall that conservative state transition matrices have, for each column, entries that sum to 1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![4. Is There A Steady State?
So far, we've seen that for a conservative state transition matrix A, we can find the eigenvector, v, corre-
sponding to the eigenvalue 2 = 1. This vector is the steady state since Av = v. However, we've so far taken
for granted that the state transition matrix even has the eigenvalue 1 = 1. Let's try to prove this fact.
(a) Show that if 2 is an eigenvalue of a matrix A, then it is also an eigenvalue of the matrix A".
Hint: The determinants of A and A' are the same. This is because the volumes which these matrices
represent are the same.
(b) Let a square matrix A have, for each row, entries that sum to one. Show that 1 = [1 1
an eigenvector of A. What is the corresponding eigenvalue?
1]" is
...
(c) Let's put it together now. From the previous two parts, show that any conservative state transition
matrix will have the eigenvalue 1 = 1. Recall that conservative state transition matrices have, for each
column, entries that sum to 1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda4a7445-4cc9-4690-867b-b0862e018c2b%2Fe38408cc-1ce4-46ff-964e-26cb782937bc%2Fxc3kq43_processed.png&w=3840&q=75)
Transcribed Image Text:4. Is There A Steady State?
So far, we've seen that for a conservative state transition matrix A, we can find the eigenvector, v, corre-
sponding to the eigenvalue 2 = 1. This vector is the steady state since Av = v. However, we've so far taken
for granted that the state transition matrix even has the eigenvalue 1 = 1. Let's try to prove this fact.
(a) Show that if 2 is an eigenvalue of a matrix A, then it is also an eigenvalue of the matrix A".
Hint: The determinants of A and A' are the same. This is because the volumes which these matrices
represent are the same.
(b) Let a square matrix A have, for each row, entries that sum to one. Show that 1 = [1 1
an eigenvector of A. What is the corresponding eigenvalue?
1]" is
...
(c) Let's put it together now. From the previous two parts, show that any conservative state transition
matrix will have the eigenvalue 1 = 1. Recall that conservative state transition matrices have, for each
column, entries that sum to 1.
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