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I'm trying to find if there is a restriction in tr(A) or eigenvalues sum for a non-negative, irreducible square matrix A. As an additional information, the row sums and the order of the matrix is given. It seems that my A matrix is also diagonal dominant.

Particularly, I'm searching for a restriction on the mean of the eigenvalues given that I only know the mean of the rows of A and that the sum of the rows are less than 1.

In case there isn't any constraint, under what circumstances or additional assumptions does the tr(A) or mean value of eigenvalues will be constraint?

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