Does this sequence converge to zero? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T04:27:19Z http://mathoverflow.net/feeds/question/107739 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107739/does-this-sequence-converge-to-zero Does this sequence converge to zero? Zhang Changhe 2012-09-21T05:56:01Z 2012-09-21T11:16:40Z <h2>Description</h2> <p>Let $\{e_n\}$, $e_n\in \mathbb{R}^p$ be a sequence of vectors, $\{U_n\}$, $U_n\in\mathbb{C}^{p\times p}$ be a sequence of <strong>unitary matrices</strong> (that is $U_i^*=U_i^{-1}$, $^*$denonts conjugate transpose). $Q\in\mathbb{R}^{p\times p}$ is a diagonal matrix with positive diagonal entries. They satisfy the following properties:</p> <p>$||QU_1e_2||_2&lt;\rho||QU_1e_1||_2$,</p> <p>$||QU_2e_3||_2&lt;\rho||QU_2e_2||_2$,</p> <p>$||QU_3e_4||_2&lt;\rho||QU_3e_3||_2$,</p> <p>...........</p> <p>$||QU_ie_{i+1}||_2&lt;\rho||QU_ie_i||_2$,...., where $\rho\in(0,1)$, $||\cdot||_2$ denotes $l_2$ norm on vectors.</p> <h2>Questions</h2> <p><strong>Does $\{e_n\}$ converge to zero?</strong> </p> <p>Some observations: consider a special case where $U_i=I$, then $e_n\rightarrow 0$. That means $\{e_n\}$ converges to zero under certain condition. I wonder if it converges when $\{U_n\}$ is an arbitrary sequence of unitary matrices.</p> http://mathoverflow.net/questions/107739/does-this-sequence-converge-to-zero/107747#107747 Answer by anton for Does this sequence converge to zero? anton 2012-09-21T07:22:03Z 2012-09-21T11:16:40Z <p>Here is a counterexample: Let $p=2$ and let $Q$ be the diagonal matrix with entries $(2,1)$. If $n$ is odd, let $e_n$ be the first standard basis vector $(1,0)^t$ and if $n$ is even, the second $(0,1)^t$. If $n$ is odd, let $U_n$ be the identity matrix and if $n$ is even, let $U_n$ be the matrix with first row (0,1) and second row (1,0), so $U_n$ just interchanges the two standard basis vectors.</p>