Conditions on negative definiteness

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Discussion Overview

The discussion revolves around conditions on a real-valued symmetric positive definite matrix ##R## that would ensure a specific relationship between two matrix inequalities involving matrices ##A, B, C,## and ##D##. The focus is on exploring the implications of these conditions in the context of linear matrix inequalities (LMIs).

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant inquires about literature addressing conditions under which a symmetric positive definite matrix ##R## guarantees that one matrix inequality implies another.
  • Another participant suggests considering Schur complements, though they express uncertainty about its applicability.
  • A later reply argues that Schur complements may not be useful due to the linear nature of the matrix inequality and the complications introduced by off-diagonal terms.
  • There is a discussion regarding the square nature of matrix ##R##, with one participant questioning the consistency of its properties as positive definite if it is not square.
  • One participant proposes that if the first matrix inequality holds, then setting ##R## as the identity matrix could satisfy the second inequality.
  • Another participant acknowledges a mistake regarding the assumptions about the properties of ##R## and suggests exploring whether matrices ##A## or ##D## can be assumed to be positive definite.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of Schur complements and the properties of matrix ##R##, indicating that the discussion remains unresolved with multiple competing perspectives.

Contextual Notes

There are limitations regarding the assumptions about the dimensions and properties of the matrices involved, particularly concerning the square nature of ##R## and the implications of its positive definiteness.

p4wp4w
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Hello, I want to know if there exist any result in literature that answers my question:
Under which conditions on the real valued matrix ## R ## (symmetric positive definite), the first argument results in and guarantees the second one:
1) for real valued matrices ##A, B, C,## and ## D ## with appropriate dimensions and ## A ## and ## D ## being symmetric:
##X=
\begin{pmatrix}
A & B+RC\\
B^T+C^TR & D\\
\end{pmatrix} < 0##
2)
##
Y
=
\begin{pmatrix}
A & B+C\\
B^T+C^T & D\\
\end{pmatrix} < 0##
Congruence transformation doesn't help since it will affect the diagonal elements as well.
Thank you all in advance.
 
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Have you tried looking at Schur complements? I'm not sure it would help, but maybe.
 
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perplexabot said:
Have you tried looking at Schur complements? I'm not sure it would help, but maybe.
No, it won't help really. The matrix inequality is already linear (LMI) and Schur complement in that sense will just cause a second problem that the off-diagonal terms will cause nonlinearty (later ##R## should be found by interior point method). I am expecting the answer to be in the form of a second LMI on ##R## but the problem is that a lot of simplifications or even assumptions can not be done since ##R## is not square.
 
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p4wp4w said:
No, it won't help really. The matrix inequality is already linear (LMI) and Schur complement in that sense will just cause a second problem that the off-diagonal terms will cause nonlinearty (later ##R## should be found by interior point method). I am expecting the answer to be in the form of a second LMI on ##R## but the problem is that a lot of simplifications or even assumptions can not be done since ##R## is not square.
Hmmm. Maybe you are right, Schur may not be of use. I was just throwing things out there.

Wait, you say in your last post that R is not square but in your original post you say it is positive definite?! Which one is it?
If R is positive definite and if you assume the first argument (1) is true, then one condition on R that results in argument (2) being satisfied is if R is simply the Identity matrix : P
 
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perplexabot said:
Hmmm. Maybe you are right, Schur may not be of use. I was just throwing things out there.

Wait, you say in your last post that R is not square but in your original post you say it is positive definite?! Which one is it?
If R is positive definite and if you assume the first argument (1) is true, then one condition on R that results in argument (2) being satisfied is if R is simply the Identity matrix : P
That's a nasty mistake that I made; I think deep down, I was looking for something like SPDness of ##R## to simplify things but ##R## is not square in general.
 
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p4wp4w said:
That's a nasty mistake that I made; I think deep down, I was looking for something like SPDness of ##R## to simplify things but ##R## is not square in general.
Can you assume ##A## or ##D## to be positive definite?
 

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