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Non-Negative irreducible matrices with random (correlated or independent) non-zero entries and deterministic zeros

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Lets $M$ be a non-negative irreducible matrix. According to Perron-Frobenius Theorem, the maximum eigenvalue of $M$, $lambda$, is positive and equal to its spectral radius $rho(M)$.

My question is that what will happen to the expected maximum eigenvalue if, only, the nonzero (positive) entries of $M$ are random with a positive distribution? For example a Gaussian with positive mean. I also want to know the effect of correlation among these entries on the maximum eigenvalue.

My observation is that when the $tanh(x)$ with $xsim N(m, sigma^2)$ is used as the non-zero elements of $M$, existence of positive correlation among the non-zero entries increases the expected maximum eigenvalue compared to the case where the entries are independent. But I ma not able to justify this experiment.

linear-algebra eigenvalues-eigenvectors random-matrices spectral-radius

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asked Mar 15 at 17:27

afraafra

63

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0

$begingroup$

Lets $M$ be a non-negative irreducible matrix. According to Perron-Frobenius Theorem, the maximum eigenvalue of $M$, $lambda$, is positive and equal to its spectral radius $rho(M)$.

My question is that what will happen to the expected maximum eigenvalue if, only, the nonzero (positive) entries of $M$ are random with a positive distribution? For example a Gaussian with positive mean. I also want to know the effect of correlation among these entries on the maximum eigenvalue.

My observation is that when the $tanh(x)$ with $xsim N(m, sigma^2)$ is used as the non-zero elements of $M$, existence of positive correlation among the non-zero entries increases the expected maximum eigenvalue compared to the case where the entries are independent. But I ma not able to justify this experiment.

linear-algebra eigenvalues-eigenvectors random-matrices spectral-radius

share|cite|improve this question

asked Mar 15 at 17:27

afraafra

63

$endgroup$

add a comment | 

$begingroup$

Lets $M$ be a non-negative irreducible matrix. According to Perron-Frobenius Theorem, the maximum eigenvalue of $M$, $lambda$, is positive and equal to its spectral radius $rho(M)$.

My question is that what will happen to the expected maximum eigenvalue if, only, the nonzero (positive) entries of $M$ are random with a positive distribution? For example a Gaussian with positive mean. I also want to know the effect of correlation among these entries on the maximum eigenvalue.

My observation is that when the $tanh(x)$ with $xsim N(m, sigma^2)$ is used as the non-zero elements of $M$, existence of positive correlation among the non-zero entries increases the expected maximum eigenvalue compared to the case where the entries are independent. But I ma not able to justify this experiment.

linear-algebra eigenvalues-eigenvectors random-matrices spectral-radius

share|cite|improve this question

asked Mar 15 at 17:27

afraafra

63

$endgroup$

share|cite|improve this question

asked Mar 15 at 17:27

afraafra

63

share|cite|improve this question

asked Mar 15 at 17:27

afraafra

63

asked Mar 15 at 17:27

afraafra

63

asked Mar 15 at 17:27

afraafra

63