most recent 30 from http://mathoverflow.net 2013-05-25T13:37:57Z http://mathoverflow.net/feeds/user/29691 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116054/signal-model-classification-between-two-possbile-candidates Soup 2012-12-11T07:35:52Z 2013-02-21T21:22:00Z

<p>How to decide the most possible signal model between two model candidates besed on the received signal vector?</p> <p>Assume the received signal vector is $y$, the possible signal model candidates could be:</p> <p>(1) $y = Ax+n$, or</p> <p>(2) $y = Bx+n$,</p> <p>in which $x$ is the transmitted signal vector, and $A$ and $B$ are the system matrices for signal model candidate-1 and candidate-2 respectively, and $n$ is the Gaussian noise vector.</p> <p>If $y$,$A$ and $B$ are all known, and the noise covariance matrix is $E[nn^H] = w^2I$, in which $w^2$ stands for noise power, and $I$ is the identity matrix, how to decide the most possible signal model candidate between the two. What's the optimal solution?</p> <p>Thanks for any discussions.</p>

http://mathoverflow.net/questions/116054/signal-model-classification-between-two-possbile-candidates/116301#116301 Soup 2012-12-13T17:24:57Z 2012-12-13T17:24:57Z

<p>A good reference book could be</p> <p>Hero, A. “Signal Detection and Classification” Digital Signal Processing Handbook Ed. Vijay K. Madisetti and Douglas B. Williams Boca Raton: CRC Press LLC, 1999</p> <p>in which different scenarios were discussed. Besides, some reference books are also helpful.</p>

http://mathoverflow.net/questions/115476/how-to-calculate-the-inverse-of-the-sum-of-two-eigen-decomposed-matrices Soup 2012-12-05T07:34:45Z 2012-12-10T19:02:57Z

<p>The are two eigen-decomposed matrices $A$ = $U_1$$V_1$$U_1$$^H$, $B$ = $U_2$$V_2$$U_2$$^H$, in which $V_1$ and $V_2$ are the eigen-matrices formed by the non-negative eigenvalues and the eigenvalues are all less than 1, $U_1$ and $U_2$ are unitary matrices formed by the eigenvectors. Is there any efficient way (including any efficient iterative solution) to calcualte the following vector?</p> <p>$y$ = ($A$+$B$+$I$)$^{-1}$$x$</p> <p>in which $I$ is the identity matrix, and $x$ could be an arbitrary vector.</p> <p>Thanks for any discussions.</p>

http://mathoverflow.net/questions/115476/how-to-calculate-the-inverse-of-the-sum-of-two-eigen-decomposed-matrices/116008#116008 Soup 2012-12-11T07:08:28Z 2012-12-11T07:08:28Z

So we only need to inverse a matrix with the size = rank($B$), right? This is a good solution. Thank you very much!

http://mathoverflow.net/questions/115476/how-to-calculate-the-inverse-of-the-sum-of-two-eigen-decomposed-matrices Soup 2012-12-07T17:32:19Z 2012-12-07T17:32:19Z

What if $A$ or $B$ have very low rank (but do not share a large eigenspace)? Any solution for this scenario? Thanks.

http://mathoverflow.net/questions/115476/how-to-calculate-the-inverse-of-the-sum-of-two-eigen-decomposed-matrices Soup 2012-12-07T06:42:04Z 2012-12-07T06:42:04Z

You are right. x is a given vector. U1 and U2 are the unitary matrices, while V1 and V2 are diagonal matrices, formed by the eigen-vectors of A and B respectively.