User anonymous coward - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T03:08:27Z http://mathoverflow.net/feeds/user/19237 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/80863/stochastic-matrix-second-largest-eigenvalue-and-second-largest-absolute-value-of Stochastic Matrix: Second largest eigenvalue and second largest absolute value of eigen value anonymous coward 2011-11-14T03:41:42Z 2011-11-14T06:14:15Z <h2>Setup</h2> <p>Let $A$ be a stochastic matrix.</p> <p>Let the eigenvalues of $A$ be $1 = \lambda_1 \geq \lambda_2 \geq \lambda_3 ... \geq -1$.</p> <p>Let $\lambda = \max_{x: x \perp 1} \frac{||Ax||}{|| x ||}$</p> <h2>Question:</h2> <p>Besides $\lambda_2 \leq \lambda$, is there any relation between $\lambda$ and $\lambda_2$? In particular, I would love to see something of the form $\lambda \leq \lambda_2$.</p> <h2>Context:</h2> <p>Reading about expanders. Many of the proofs appears to prove upper bounds on $\lambda_2$, but I want upper bounds on $\lambda$, and it's not obvious to me:</p> <p>(1) how an upper bound on $\lambda_2$ becomes an upper bound on $\lambda$ or (2) how to generalize some of these proofs.</p> <p>Thanks!</p>