Discussion on the eigenvalues of the coupling matrix and the direct product of matrices.

Conversation started Aug 31, 2017 at 13:22.

BAYMAX

Aug 31, 2017 13:22

*)If $G$ is a real symmetric matrix with the row sums zero then the matrix is diagonalizable ?

Also how the zero row sum condition can lead us to have a spectrum of eigenvalues of matrix $G$ to be entirely semipositive that is $\lambda_{i} \geq 0 \forall i$

18 hours later…

BAYMAX

Sep 1, 2017 07:21

@MartinSleziak just in case if you see this question in your free time.Just wanted that you may have a look at the question!

Martin Sleziak

@BAYMAX isn't every real symmetric matrix diagonalizable?

The second question about row sums and eigenvalues seems more interesting.

BAYMAX

Hmm the first one is a fact!

Martin Sleziak

But I don't think the thing about zero sum condition is true. Where does this claim come from?

Notice that if $A$ has zero row-sums so does $-A$. And the eigenvalues of these two matrices are exactly opposite.

BAYMAX

its from a dynamical systems book.

will add the screen shot

Martin Sleziak

Maybe you can also tell me the name of the book?

BAYMAX

Sep 1, 2017 07:29

"Synchronized Dynamics of Complex systems volume 6"

page 21

Martin Sleziak

> Further, because the zero row condition, the spectrum of eigenvalues is entirely semi-positive, ...

BAYMAX

Yes

Martin Sleziak

To be honest, I do not see a flaw in my argument above explaining why this is not ture... :-(

BAYMAX

yes,I too see the logic there

oh,wai

Martin Sleziak

Sep 1, 2017 07:46

Was this supposed to by "oy, vey" or "oh, wait"? :-)

Anyway, I'll have to go. Let's hope somebody who knows about this stuff notices your question.

BAYMAX

I thought I got the logic but then it was wrong and i typed it so :)

BAYMAX

Sep 1, 2017 08:11

Oh,wai - "Actually if we see I think the elements of the coupling matrix $G$ must be positive so even if $-G$ is not physically feasible!,so .. "

so our question reduces to this if $G$ is a symmetric matrix with positive entries then

how the zero row sum condition can lead us to have a spectrum of eigenvalues of matrix $G$ to be entirely semipositive that is $\lambda_{i} \geq 0 \forall i$

@MartinSleziak

again i got caught in mistake

if the entries of the matrix are positive

then how the row sum will be 0

only possible when the matrix is zero matrix

sorry for ping

Martin Sleziak

In the Wikipedia article on diagonally dominant matrices I see that such a matrix has non-negative eigenvalues by Gershgorin circle theorem. (Assuming the eigenvalues are real.)

The definition of diagonally dominant is $|a_{ii}| \geq \sum\limits_{j\neq i} |a_{ij}|.$

From zero-sum condition you know that $-a_{ii} = \sum\limits_{j\neq i} a_{ij}$.

So if non-diagonal entries of your coupling matrix $G$ have the same sign, then you get non-negative eigenvalues.

BAYMAX

ye I got that in our case of zero sum condition $|a_{ii}| = \sum_{j \neq i} |a_{ij}|$

and hence from Gershgorin theorem we get that the matrix has non-negative eigenvalues

but how you concluded that the non-diagonal entries of the matrix have the same sign?

Martin Sleziak

Sep 1, 2017 09:22

@BAYMAX No, I did not conclude that they have the same sign, I just asked whether this would be a reasonable additional assumption.

It seems that in Boccaletti's there is some assumption about the coupling matrix $G$ which we are missing.

Maybe the author forgot to mention the additional condition. Maybe some additional condition follows from some properties of the systems studied there.

BAYMAX

oh,what is that assumption page no ?

Martin Sleziak

I do not know whether such assumption is in the book.

BAYMAX

Like?

Martin Sleziak

But similarly as you guessed that there might be additional assumption that all entries are positive, I guessed that we might have this additional assumption.

@BAYMAX I do not understand what you are asking in "Like?"

BAYMAX

like which assumption which we are missing!

but all entries

positive

leads to the problem of having row sum zero

Martin Sleziak

Sep 1, 2017 09:26

Yes.

You said just a while ago this: "I think the elements of the coupling matrix $G$ must be positive so even if $-G$ is not physically feasible!"

BAYMAX

yes

may be I am missing something.

Martin Sleziak

So you added some condition on $G$ (positive entries) not based on something in section 2.6 but based on something you know about the type problems studied there. (Based on what is "physically feasible" - I do not know what is physical feasible because I do not even know what kind of physical problem the matrix $G$ should model.)

So what I was trying to say above is - maybe from physical interpretation of $G$ you can get that non-diagonal elements have the same sign.

BAYMAX

hmm,I see, I have to dig a bit more!

also a "direct product of matrices?"

Martin Sleziak

Could it be Kronecker product?

BAYMAX

returned nothing

Martin Sleziak

Sep 1, 2017 09:33

I gave you Wikipedia link in the previous message.

BAYMAX

yes i see the same symbol as in eqn 2.10

yes

Martin Sleziak

Same symbol does not necessarily mean the same thing, but maybe from the context it is possible to judge whether this is what the author of the book meant.

BAYMAX

he mentioned it to be direct product

but as i see there are Kronecker product , outer product

in wiki

Martin Sleziak

Sep 1, 2017 09:48

There is also this question on the main site: A confusion on terminologies: “direct product” and “tensor product” of matrices. But to me the question itself seems a bit unclear.

Conversation ended Sep 1, 2017 at 9:48.

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