I am seeking assistance in investigating the following result.
Assume matrix $A \in \mathbb{R}^{n \times n}$ a nonnegative, irreducible and row-stochastic matrix. I wonder if the 2-norm $\|(I_n - \frac{\bf{1} \bf{1}^T}{n}) A\|_2 < 1$ holds? The matrix $I_n$ is the identical matrix, and $\bf{1}$ is the column vector with all ones.
For the above matrix $A$, there are several common properties that may be useful in the proof: $A\bf{1} = \bf{1}$ and the eigenvalues $1 = \lambda_{\max} (A) > \lambda_2 \geq \dots \geq \lambda_n \geq 0$
This norm can be used to characterize the convergence rate of the consensus over a directed network, which is useful in the fields of distributed networks and computer science. Note that for the special cases when matrix $A$ is doubly stochastic and symmetric, the result $\|(I_n - \frac{\bf{1} \bf{1}^T}{n}) A\|_2 < 1$, which is the second largest eigenvalue that reflects the convergence rate of $A^t$ to $\frac{\bf{1} \bf{1}^T}{n}$.
Thanks.