Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Given $A$ a matrix with spectral radius smaller than 1 and a symmetric matrix $C$. It can be shown that $U=\sum_{k=0}^\infty (A^T)^k C A^k$ converges, is symmetric and is the solution of the equation above.

Is it possible to show that if $C$ is non-negative also $U$ is non-negative?

share|cite|improve this question

For every $x$, $x^*Ux=\sum\limits_{k\geqslant0}x_k^*Cx_k$, where $x_k=A^kx $ for every $k\geqslant0$. If $C$ is nonnegative, then $x_k^*Cx_k\geqslant0$ for each $k\geqslant0$ hence $x^*Ux\geqslant0$. Thus, $U$ is nonnegative.

share|cite|improve this answer
Thanks Did. I agree that if $C$ is semidefinite positive, then also $U$ is (this is what you proved). I was asking if we can claim that if $C$ is entrywise non-negative then $U$ is entrywise non-negative. Now I doubt this is true but I haven't found a counter-example. – Giovanni Aug 28 '12 at 8:59

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.