Hi everyone,
I have a matrix defined such that each row has a sum lesser than one. I want to know if there is a theorem or result saying that the spectral radius of these kind of matrices is always lesser than one.
Using simulation, I verified that this result holds (1000000 matrices simulated randomly such that the sum of each line is 0.99999) but I'm looking for a theoretical result.
Best regards,
r(A) denote spectral radius of a matrix A, |A| denote the row sum norm of A, it is known that r(A) is less than or equal to any vector induced matrix norm of A, the row sum norm is a particular type of vector induced matrix norm.
If the matrix A you considering also satisfy the condition that absolute sum of each of the rows(sum of absolute values of each row elements) is less than or equal to 1, then row sum norm of A is less than or equal to 1, thus from above restriction relating r(A) and |A|, r(A) is also less than or equal to one.
However, if the row sum of each rows is less than or equal to 1, but absolute sum for at least one row exceed 1, row sum norm becomes more than one, and restriction on r(A) to be less than or equal to one, do not apply in this situation.
Dear Ouzzani Fares ,
In general, the answer is No.
Counterexample:
The 2x2 matrix A= [ -10 10.1
-20 20.3]
has the spectral radius is r(A) > 1.
One can prove the following proposition :
If all the entries of the matrix A are positive and the sum of each row ( or column) is less than 1, then r(A) < 1.
Best wishes
Dear Peter Breuer ,
Gershgorin disks work for columns too.
One can observe that A and AT have the same spectrum.
Best wishes
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