Avoiding zero solution for LMI for bolck matrix

[mohmmed ahmed]

assume matrix A,B,C 2X2  square matrix  and i have this LMI

 AP+A'P  BP;B^TP  C]<0 ,P>0]

now if i want to add epsilon*eye(2,2) to avoid  getting eig(P) near to zero like 2e-9

then why the epsilon*eye(2,2) should be added to the term AP+A'P only

and other  term Not except

i.e  only possible solution

[AP+A'P  BP;B^TP c]+[epsilon*eye(2,2)  0;0  0]<0]

WHY?

[Johan Löfberg]

Is someone telling you absolutely have to do it that way? I would probably do P>=scale*I (for a significant scale, not 1e-8 or some small noise like that which will get lost in the numerics of a solver)

 [AP+A'P  BP;B^TP  C]<=0 ,P>=scale*I]

or why not 

sdpvar scale

 [AP+A'P+scale*I  BP;B^TP  C]<=0 ,P>=scale*I]

or

 [AP+A'P  BP;B^TP  C]<=-scale*I ,P>=scale*I]

and then maximize scale

(except that I think you mean [AP+A'P  PB;B'P  C])

[mohmmed ahmed]

thanks johan

does

[AP+A'P  BP;B^TP C]+[epsilon*eye(2,2)  0;0  epsilon*eye(2,2)]<0]

equivelent to

[AP+A'P  BP;B^TP C] <0]

and why

[Johan Löfberg]

Depends on what you mean with epsilon.

Don't complicate the question with LMIs and blocks etc. You are asking if

x+eps <0

is the same as

x < 0

If eps is an actual finite number, never mind how small, they are trivially two different open sets. x+0.001 < 0 is obviously not the same as x<0

If eps is a infinitedecimally small number, then the two sets are the same, if I remember my real analysis correctly.