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These are replies submitted by rzarouf

@vv Thank you very much!

@sand15 many thanks!!

@acer Thank you so much! Your remarks and your code are extremely useful to me! :-D

@Carl Love Many thanks. Indeed with eps=0.1 and n=3 it works fine, see attached file built on acer's.

I would also need to name the axis, i.e put "x" and "y" (maybe you know the most convenient way to do this?) and also name the curves: 
red : "Diagonal matrix"
blue : "Non normal matrix"
green : "Model matrix".

I tried to use the command "legend = ..." inside the command "implicitplot", but when I write legend = Diagonal matrix, Maple understands "Diagonal" as a sort of command from the package LinearAlgebra and thus returns "LinearAlgebra:-Diagonal matrix" instead of "Diagonal matrix". Do you know how to avoid this ?

Download rzarouf_02.mw

Thank you very much for your letter and for your precious code. I am currently trying to understand it. In the meantime I am attaching the one wrote (see PLOTS_PSEUDO_SPECTRA.mw or below), which returns the plots I need. Do you understand at first sight why my procedure returns curves whereas yours returns dots ? 


@david2024 Many thanks for your message and for your interest in this question!

I tried to run your procedure but I receive the error message :

"Error, (in P0) annotate is not a command in the plottools package"

-> Thus I have added the package plottools, i.e. with(plottools);  but I still receive the same error message.
Is it the same for you ?  

@acer Thanks for your message. All eigenvalues lie in the closed unit disk because the random matrix N is normalized. Thus, the range x=-2..2, y=-2..2, is large enough to view the 3 level sets defining the pseudospectras. Sorry my question was not enough precise: I would like the plots to have the same SIZES. Because of course the range depends on the location of the eigenvalues. 

Sounds really gound. Many many thanks for all of your explanations. It is now crystal clear

Thanks a lot  dharr 4726 !! I tried to write a code myself (building on what you wrote and on the second definition given in the wikipedia link you sent) but that didn't work. Maybe you could help me to understand why ?
I wrote : 

A := LinearAlgebra:-DiagonalMatrix([-1., I + 1., -I]);
implicitplot((0.1)^(-1) < Norm(1/((x + y*I)*IdentityMatrix(3) - A), 2), x = -1 .. 1, y = -1 .. 1);

Error, (in factor/remember) floating point coefficients are incompatible with field extension; use 'real' or 'complex' instead
implicitplot((0.1)^(-1) < 1/min(SingularValues((x + y*I)*IdentityMatrix(3) - A)), x = -1 .. 1, y = -1 .. 1);
Error, (in LinearAlgebra:-Eigenvalues) expecting either Matrices of rationals, rational functions, radical functions, algebraic numbers, or algebraic functions, or Matrices of complex(numeric) values
I don't understand why none of these commands work. The error message is red colored above.
Many thanks once again


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