Timeline for On solution of a class of discrete-time Lyapunov equation
Current License: CC BY-SA 3.0
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| May 8, 2013 at 19:41 | comment | added | Suvrit | Also note that if $\|\sum_i F_i \kron F_i\| < 1$, then starting from $X_0=I$, you can iterate $X_{k+1} = \sum_i F_iX_kF_i^T + C$, and converge to the unique semidefinite solution. If the operators don't satisfy this sufficient condition, then more thought is needed. | |
| May 8, 2013 at 17:59 | comment | added | Suvrit | Ah, so you are asking whether $\text{mat}(x)$ will be positive semidefinite? Well, if the original linear system has a unique solution that is guaranteed to be semidefinite, then by solving the vectorized linear system, we should end up recovering that. It remains to determine conditions under which the original system has a semidefinite solution. The above system seems to me to be a well-studied problem; hopefully, F. Poloni (on MO) takes a look and provides some more details, as I think he knows more about these things. | |
| May 8, 2013 at 12:04 | comment | added | eolithr | Thanks. Dr. S. Sra for your kind help. Indeed, we have thought of the vectorization method. But we still have a question that does the solution obtained by vectorization method has a 'clear' or the 'same' meaning as the original one. For example, the original solution means a covariance of some statistical variables, but when we use the vectorization method, we may collect up the vectors to build a 'suitable' solution, does this solution still means the covariance? | |
| May 8, 2013 at 11:55 | vote | accept | eolithr | ||
| May 7, 2013 at 18:17 | history | answered | Suvrit | CC BY-SA 3.0 |