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1

$begingroup$

I have the following $4n times 4n$ block tridiagonal matrix:

$$
beginbmatrix
M_1 && -M_2 && 0 && cdots&&&& &&0 \
M_2 && M_1 && -JI_2 &&cdots&&&&&& 0\
0&& JI_2 && 0 && -hI_2 &&&& && vdots\
0&& 0 && hI_2 && 0 &&ddots&& &&\
vdots && &&&& ddots&&ddots && -JI_2 && 0\
0 && && &&&&JI_2&& M_1^'&&-M_2^'\
0 && 0 &&&& cdots && 0&&M_2^' &&M_1^'
endbmatrix
$$

Each block is a $2times2$ matrix with complex entries and $I_2$ is the $2times2$ identity matrix and $0neq h,J in mathbbR$ and I know the following:

1. $M_1,M_1^'$ are diagonal matrices


2. $M_2$ and $M_2^'$ are of the form $beginbmatrix h && * \ 0 && hendbmatrix$

My question is: Can we compute the eigenvalues and eigenvectors exactly knowing this information?

linear-algebra matrices eigenvalues-eigenvectors block-matrices tridiagonal-matrices

share|cite|improve this question

edited Mar 16 at 8:06

Rodrigo de Azevedo

13.2k41960

asked Feb 24 '18 at 22:22

user1058860user1058860

285111

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What is the matrix $J$ and what is $h$?
  $endgroup$
  – JimmyK4542
  Feb 25 '18 at 3:34
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  sorry, they are real numbers. made the change
  $endgroup$
  – user1058860
  Feb 25 '18 at 3:39
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Its eigenvalues are real . . . en.wikipedia.org/wiki/Tridiagonal_matrix
  $endgroup$
  – Chickenmancer
  Feb 25 '18 at 4:11
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  where did you get that from?
  $endgroup$
  – user1058860
  Feb 25 '18 at 4:15
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Also, what is the pattern of $JI_2$ and $hI_2$ blocks? Is it one $JI_2$ block, followed by $4n-5$ $hI_2$ blocks, followed by one $JI_2$ block, or is the pattern something else?
  $endgroup$
  – JimmyK4542
  Feb 25 '18 at 4:58
  

  
  
  
  

 | 
show 1 more comment

1

$begingroup$

I have the following $4n times 4n$ block tridiagonal matrix:

$$
beginbmatrix
M_1 && -M_2 && 0 && cdots&&&& &&0 \
M_2 && M_1 && -JI_2 &&cdots&&&&&& 0\
0&& JI_2 && 0 && -hI_2 &&&& && vdots\
0&& 0 && hI_2 && 0 &&ddots&& &&\
vdots && &&&& ddots&&ddots && -JI_2 && 0\
0 && && &&&&JI_2&& M_1^'&&-M_2^'\
0 && 0 &&&& cdots && 0&&M_2^' &&M_1^'
endbmatrix
$$

Each block is a $2times2$ matrix with complex entries and $I_2$ is the $2times2$ identity matrix and $0neq h,J in mathbbR$ and I know the following:

1. $M_1,M_1^'$ are diagonal matrices


2. $M_2$ and $M_2^'$ are of the form $beginbmatrix h && * \ 0 && hendbmatrix$

My question is: Can we compute the eigenvalues and eigenvectors exactly knowing this information?

linear-algebra matrices eigenvalues-eigenvectors block-matrices tridiagonal-matrices

share|cite|improve this question

edited Mar 16 at 8:06

Rodrigo de Azevedo

13.2k41960

asked Feb 24 '18 at 22:22

user1058860user1058860

285111

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What is the matrix $J$ and what is $h$?
  $endgroup$
  – JimmyK4542
  Feb 25 '18 at 3:34
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  sorry, they are real numbers. made the change
  $endgroup$
  – user1058860
  Feb 25 '18 at 3:39
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Its eigenvalues are real . . . en.wikipedia.org/wiki/Tridiagonal_matrix
  $endgroup$
  – Chickenmancer
  Feb 25 '18 at 4:11
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  where did you get that from?
  $endgroup$
  – user1058860
  Feb 25 '18 at 4:15
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Also, what is the pattern of $JI_2$ and $hI_2$ blocks? Is it one $JI_2$ block, followed by $4n-5$ $hI_2$ blocks, followed by one $JI_2$ block, or is the pattern something else?
  $endgroup$
  – JimmyK4542
  Feb 25 '18 at 4:58
  

  
  
  
  

 | 
show 1 more comment

$begingroup$

I have the following $4n times 4n$ block tridiagonal matrix:

$$
beginbmatrix
M_1 && -M_2 && 0 && cdots&&&& &&0 \
M_2 && M_1 && -JI_2 &&cdots&&&&&& 0\
0&& JI_2 && 0 && -hI_2 &&&& && vdots\
0&& 0 && hI_2 && 0 &&ddots&& &&\
vdots && &&&& ddots&&ddots && -JI_2 && 0\
0 && && &&&&JI_2&& M_1^'&&-M_2^'\
0 && 0 &&&& cdots && 0&&M_2^' &&M_1^'
endbmatrix
$$

Each block is a $2times2$ matrix with complex entries and $I_2$ is the $2times2$ identity matrix and $0neq h,J in mathbbR$ and I know the following:

1. $M_1,M_1^'$ are diagonal matrices


2. $M_2$ and $M_2^'$ are of the form $beginbmatrix h && * \ 0 && hendbmatrix$

My question is: Can we compute the eigenvalues and eigenvectors exactly knowing this information?

linear-algebra matrices eigenvalues-eigenvectors block-matrices tridiagonal-matrices

share|cite|improve this question

edited Mar 16 at 8:06

Rodrigo de Azevedo

13.2k41960

asked Feb 24 '18 at 22:22

user1058860user1058860

285111

$endgroup$

share|cite|improve this question

edited Mar 16 at 8:06

Rodrigo de Azevedo

13.2k41960

asked Feb 24 '18 at 22:22

user1058860user1058860

285111

share|cite|improve this question

edited Mar 16 at 8:06

Rodrigo de Azevedo

13.2k41960

asked Feb 24 '18 at 22:22

user1058860user1058860

285111

edited Mar 16 at 8:06

Rodrigo de Azevedo

13.2k41960

edited Mar 16 at 8:06

Rodrigo de Azevedo

13.2k41960

asked Feb 24 '18 at 22:22

user1058860user1058860

285111

asked Feb 24 '18 at 22:22

user1058860user1058860

285111