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How to write this inequality in terms of Schur Complement?

0

$begingroup$

I know the basis about Schur-Complement. Anyway, while looking at this inequality to apply it in order to solve for $lambda$ such that the the matrix is definite positive, I got a little bit confused because the lack of non inverse terms.

$ X ^ T left( PA+ A ^ T P right) X<-lambda X ^ T AX$

May I... ?

$beginbmatrix 0 & X \ X ^ T & left( PA+ A ^ T P+lambda A right) ^ -1 endbmatrix$

Or,

$beginbmatrix X ^ T left( PA+ A ^ T P+lambda A right) X & P \ P & 0 endbmatrix$

Or even,

$beginbmatrix X ^ T left( PA+ A ^ T P right) X & X \ X ^ T & -lambda A ^ -1 endbmatrix$

I'm sorry if this is a silly or even stupid question, but i'm lost :(
Any references, books, or examples that You could recommend me?

lmis schur-complement

share|cite|improve this question

asked Apr 1 at 5:58

Francisco MendozaFrancisco Mendoza

1

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  1) Which matrix should be definite positive ? 2) What are the dimensions of your matrices ?
  $endgroup$
  – Jean Marie
  Apr 1 at 9:37
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Could you answer my questions and/or say if my answer brings something to you ?
  $endgroup$
  – Jean Marie
  Apr 1 at 17:33
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The only thing my professor gave us is A, a 3x3 matrix, and that $x^T·x=||x||^2$ :( A'm sorry if it's pretty ambiguous :( The mean objective to use S-Complement, is to use YALMIP over MATLAB to solve for $lambda$; at least that's what I understood. If you need more information, please list it, and I will ask my professor if there is any missing information. Thanks in advance!
  $endgroup$
  – Francisco Mendoza
  Apr 1 at 19:26
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Is $lambda$ your only unknown because that is the only variable you want to solve for? If that is the case then your first equation is already a LMI.
  $endgroup$
  – Kwin van der Veen
  Apr 3 at 15:15
  
  

  
  
  
  

add a comment | 

0

$begingroup$

I know the basis about Schur-Complement. Anyway, while looking at this inequality to apply it in order to solve for $lambda$ such that the the matrix is definite positive, I got a little bit confused because the lack of non inverse terms.

$ X ^ T left( PA+ A ^ T P right) X<-lambda X ^ T AX$

May I... ?

$beginbmatrix 0 & X \ X ^ T & left( PA+ A ^ T P+lambda A right) ^ -1 endbmatrix$

Or,

$beginbmatrix X ^ T left( PA+ A ^ T P+lambda A right) X & P \ P & 0 endbmatrix$

Or even,

$beginbmatrix X ^ T left( PA+ A ^ T P right) X & X \ X ^ T & -lambda A ^ -1 endbmatrix$

I'm sorry if this is a silly or even stupid question, but i'm lost :(
Any references, books, or examples that You could recommend me?

lmis schur-complement

share|cite|improve this question

asked Apr 1 at 5:58

Francisco MendozaFrancisco Mendoza

1

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  1) Which matrix should be definite positive ? 2) What are the dimensions of your matrices ?
  $endgroup$
  – Jean Marie
  Apr 1 at 9:37
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Could you answer my questions and/or say if my answer brings something to you ?
  $endgroup$
  – Jean Marie
  Apr 1 at 17:33
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The only thing my professor gave us is A, a 3x3 matrix, and that $x^T·x=||x||^2$ :( A'm sorry if it's pretty ambiguous :( The mean objective to use S-Complement, is to use YALMIP over MATLAB to solve for $lambda$; at least that's what I understood. If you need more information, please list it, and I will ask my professor if there is any missing information. Thanks in advance!
  $endgroup$
  – Francisco Mendoza
  Apr 1 at 19:26
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Is $lambda$ your only unknown because that is the only variable you want to solve for? If that is the case then your first equation is already a LMI.
  $endgroup$
  – Kwin van der Veen
  Apr 3 at 15:15
  
  

  
  
  
  

add a comment | 

$begingroup$

I know the basis about Schur-Complement. Anyway, while looking at this inequality to apply it in order to solve for $lambda$ such that the the matrix is definite positive, I got a little bit confused because the lack of non inverse terms.

$ X ^ T left( PA+ A ^ T P right) X<-lambda X ^ T AX$

May I... ?

$beginbmatrix 0 & X \ X ^ T & left( PA+ A ^ T P+lambda A right) ^ -1 endbmatrix$

Or,

$beginbmatrix X ^ T left( PA+ A ^ T P+lambda A right) X & P \ P & 0 endbmatrix$

Or even,

$beginbmatrix X ^ T left( PA+ A ^ T P right) X & X \ X ^ T & -lambda A ^ -1 endbmatrix$

I'm sorry if this is a silly or even stupid question, but i'm lost :(
Any references, books, or examples that You could recommend me?

lmis schur-complement

share|cite|improve this question

asked Apr 1 at 5:58

Francisco MendozaFrancisco Mendoza

1

$endgroup$

share|cite|improve this question

asked Apr 1 at 5:58

Francisco MendozaFrancisco Mendoza

1

share|cite|improve this question

asked Apr 1 at 5:58

Francisco MendozaFrancisco Mendoza

1

asked Apr 1 at 5:58

Francisco MendozaFrancisco Mendoza

1

asked Apr 1 at 5:58

Francisco MendozaFrancisco Mendoza

1

1 Answer
1

active

oldest

votes

0

$begingroup$

I don't see the point of using Schur's complements here.

In fact, looking for $lambda$ satisfying inequality

$$X^Tleft( PA+A^TP right) X<-lambda X^TAXtag1$$

is the same as looking for the bounds of

$$dfracX^TBXX^TAX textwhere B:=PA+A^TP tag2$$

which is a classical issue :

If $A$ is positive-definite, consider a Cholesky decomposition $A=C^TC$, set $Y=CX$, transforming (2) into the Rayleigh quotient :

$$dfracYC^-TBC^-1YY^TY $$

which is known to take all values in interval $[lambda_min,lambda_max]$
where $lambda_min,lambda_max$ are the extreme eigenvalues of matrix $D:=C^-TBC^-1$.

Connected :

Proof: Ratio of matrix traces and difference of traces

Ratio of two quadratic vector forms

share|cite|improve this answer

edited Apr 1 at 17:32

answered Apr 1 at 9:35

Jean MarieJean Marie

31.6k42355

$endgroup$

add a comment | 

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0

$begingroup$

I don't see the point of using Schur's complements here.

In fact, looking for $lambda$ satisfying inequality

$$X^Tleft( PA+A^TP right) X<-lambda X^TAXtag1$$

is the same as looking for the bounds of

$$dfracX^TBXX^TAX textwhere B:=PA+A^TP tag2$$

which is a classical issue :

If $A$ is positive-definite, consider a Cholesky decomposition $A=C^TC$, set $Y=CX$, transforming (2) into the Rayleigh quotient :

$$dfracYC^-TBC^-1YY^TY $$

which is known to take all values in interval $[lambda_min,lambda_max]$
where $lambda_min,lambda_max$ are the extreme eigenvalues of matrix $D:=C^-TBC^-1$.

Connected :

Proof: Ratio of matrix traces and difference of traces

Ratio of two quadratic vector forms

share|cite|improve this answer

edited Apr 1 at 17:32

answered Apr 1 at 9:35

Jean MarieJean Marie

31.6k42355

$endgroup$

add a comment | 

0

$begingroup$

I don't see the point of using Schur's complements here.

In fact, looking for $lambda$ satisfying inequality

$$X^Tleft( PA+A^TP right) X<-lambda X^TAXtag1$$

is the same as looking for the bounds of

$$dfracX^TBXX^TAX textwhere B:=PA+A^TP tag2$$

which is a classical issue :

If $A$ is positive-definite, consider a Cholesky decomposition $A=C^TC$, set $Y=CX$, transforming (2) into the Rayleigh quotient :

$$dfracYC^-TBC^-1YY^TY $$

which is known to take all values in interval $[lambda_min,lambda_max]$
where $lambda_min,lambda_max$ are the extreme eigenvalues of matrix $D:=C^-TBC^-1$.

Connected :

Proof: Ratio of matrix traces and difference of traces

Ratio of two quadratic vector forms

share|cite|improve this answer

edited Apr 1 at 17:32

answered Apr 1 at 9:35

Jean MarieJean Marie

31.6k42355

$endgroup$

add a comment | 

$begingroup$

I don't see the point of using Schur's complements here.

In fact, looking for $lambda$ satisfying inequality

$$X^Tleft( PA+A^TP right) X<-lambda X^TAXtag1$$

is the same as looking for the bounds of

$$dfracX^TBXX^TAX textwhere B:=PA+A^TP tag2$$

which is a classical issue :

If $A$ is positive-definite, consider a Cholesky decomposition $A=C^TC$, set $Y=CX$, transforming (2) into the Rayleigh quotient :

$$dfracYC^-TBC^-1YY^TY $$

which is known to take all values in interval $[lambda_min,lambda_max]$
where $lambda_min,lambda_max$ are the extreme eigenvalues of matrix $D:=C^-TBC^-1$.

Connected :

Proof: Ratio of matrix traces and difference of traces

Ratio of two quadratic vector forms

share|cite|improve this answer

edited Apr 1 at 17:32

answered Apr 1 at 9:35

Jean MarieJean Marie

31.6k42355

$endgroup$

share|cite|improve this answer

edited Apr 1 at 17:32

answered Apr 1 at 9:35

Jean MarieJean Marie

31.6k42355

answered Apr 1 at 9:35

Jean MarieJean Marie

31.6k42355

answered Apr 1 at 9:35

Jean MarieJean Marie

31.6k42355