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eigenvector problem: solving matrices containing e and finding eigenvectors and eigenvalues

-1

$begingroup$

I have been given a matrix to solve

$S(t)$ = $1/2$ $[(e^t+e^-t),(e^t-e^-t),0,(e^t-e^-t),(e^t+e^-t),0,0,0,2]$

I am fairly sure that I have correctly calculated the first few parts of the question and have solved 3/5 questions: (** means unsolved)
Image of question

a. $det(S) = 1$

b. **Calculate the characteristic polynomial of S(t)?

c. Show $S(-t)S(t)=S(t)S(-t)= I$ //using multiplication of the matrices have solved for I

d. Using the property of S(t) derived in c., calculate $S^-1(t)$ //used the $AB = I$ rules to solve for the inverse $S^-1(t)$

e. **calculate the eigenvectors and eigenvalues of S(t)?

The two I am struggling with are $b$ and $e$, clearly calculating the characteristic polynomial is the first step of $e$; which is why I am quite stuck.

I'm sure there is a really clear way of looking at this and if anyone has any suggestions please go ahead!

matrices polynomials eigenvalues-eigenvectors

share|cite|improve this question

edited Mar 24 at 5:08

Alexander Quinn

53

asked Mar 24 at 5:01

tika-takatika-taka

1

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The characteristic polynomial is $det( lambda I - S)$. Why can't you calculate it?
  $endgroup$
  – Robert Israel
  Mar 24 at 5:15
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Well, I think I am going about it the right way but I'm not sure. Expanding from the third row I get: (2-Y)(((e^t+e^-t)-Y)^2-(e^t-e^-t)^2). Solving for those I end up getting an answer with imaginary roots which leads me to think I may have gotten something wrong in the process.
  $endgroup$
  – tika-taka
  Mar 24 at 7:48
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If you do it right, the characteristic polynomial should factor nicely. The eigenvalues are real (assuming $t$ is real).
  $endgroup$
  – Robert Israel
  Mar 24 at 19:22
  

  
  
  
  

add a comment | 

-1

$begingroup$

I have been given a matrix to solve

$S(t)$ = $1/2$ $[(e^t+e^-t),(e^t-e^-t),0,(e^t-e^-t),(e^t+e^-t),0,0,0,2]$

I am fairly sure that I have correctly calculated the first few parts of the question and have solved 3/5 questions: (** means unsolved)
Image of question

a. $det(S) = 1$

b. **Calculate the characteristic polynomial of S(t)?

c. Show $S(-t)S(t)=S(t)S(-t)= I$ //using multiplication of the matrices have solved for I

d. Using the property of S(t) derived in c., calculate $S^-1(t)$ //used the $AB = I$ rules to solve for the inverse $S^-1(t)$

e. **calculate the eigenvectors and eigenvalues of S(t)?

The two I am struggling with are $b$ and $e$, clearly calculating the characteristic polynomial is the first step of $e$; which is why I am quite stuck.

I'm sure there is a really clear way of looking at this and if anyone has any suggestions please go ahead!

matrices polynomials eigenvalues-eigenvectors

share|cite|improve this question

edited Mar 24 at 5:08

Alexander Quinn

53

asked Mar 24 at 5:01

tika-takatika-taka

1

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The characteristic polynomial is $det( lambda I - S)$. Why can't you calculate it?
  $endgroup$
  – Robert Israel
  Mar 24 at 5:15
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Well, I think I am going about it the right way but I'm not sure. Expanding from the third row I get: (2-Y)(((e^t+e^-t)-Y)^2-(e^t-e^-t)^2). Solving for those I end up getting an answer with imaginary roots which leads me to think I may have gotten something wrong in the process.
  $endgroup$
  – tika-taka
  Mar 24 at 7:48
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If you do it right, the characteristic polynomial should factor nicely. The eigenvalues are real (assuming $t$ is real).
  $endgroup$
  – Robert Israel
  Mar 24 at 19:22
  

  
  
  
  

add a comment | 

$begingroup$

I have been given a matrix to solve

$S(t)$ = $1/2$ $[(e^t+e^-t),(e^t-e^-t),0,(e^t-e^-t),(e^t+e^-t),0,0,0,2]$

I am fairly sure that I have correctly calculated the first few parts of the question and have solved 3/5 questions: (** means unsolved)
Image of question

a. $det(S) = 1$

b. **Calculate the characteristic polynomial of S(t)?

c. Show $S(-t)S(t)=S(t)S(-t)= I$ //using multiplication of the matrices have solved for I

d. Using the property of S(t) derived in c., calculate $S^-1(t)$ //used the $AB = I$ rules to solve for the inverse $S^-1(t)$

e. **calculate the eigenvectors and eigenvalues of S(t)?

The two I am struggling with are $b$ and $e$, clearly calculating the characteristic polynomial is the first step of $e$; which is why I am quite stuck.

I'm sure there is a really clear way of looking at this and if anyone has any suggestions please go ahead!

matrices polynomials eigenvalues-eigenvectors

share|cite|improve this question

edited Mar 24 at 5:08

Alexander Quinn

53

asked Mar 24 at 5:01

tika-takatika-taka

1

$endgroup$

share|cite|improve this question

edited Mar 24 at 5:08

Alexander Quinn

53

asked Mar 24 at 5:01

tika-takatika-taka

1

share|cite|improve this question

edited Mar 24 at 5:08

Alexander Quinn

53

asked Mar 24 at 5:01

tika-takatika-taka

1

edited Mar 24 at 5:08

Alexander Quinn

53

edited Mar 24 at 5:08

Alexander Quinn

53

asked Mar 24 at 5:01

tika-takatika-taka

1

asked Mar 24 at 5:01

tika-takatika-taka

1