User silvanmx - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T16:38:28Z http://mathoverflow.net/feeds/user/5066 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96016/another-matrix-diagonalization-problem Another matrix diagonalization problem silvanmx 2012-05-04T19:55:08Z 2012-05-08T15:38:04Z <p>Given the matrices $X$ and $Y$ in $[0,1]^{n\times m}$, for $n > m > 3$, so that $X1_m=1_m$ and $Y1_m=1_m$, where $1_m$ denotes a $m$-length column vector of ones, find a matrix $Q$ in $R^{m\times m}$ so that $C(X,Y)=Q^TX^TYQ$ satisfies the following tree conditions:</p> <p>$i$) $C(X,X)$ is diagonal with diagonal elements given by the the sum of rows of $X$</p> <p>$ii$) The sum of the columns of $C(X,Y)$ equals the sum of rows of $X$</p> <p>$iii$) The sum of the rows of $C(X,Y)$ equals the sum of rows of $Y$</p> <p>What can be said about the uniqueness of the matrix $Q$?</p> http://mathoverflow.net/questions/92603/a-matrix-diagonalization-problem A matrix diagonalization problem silvanmx 2012-03-29T19:40:17Z 2012-04-30T23:46:06Z <p>For matrices $X,Y\in [0,1]^{n\times m}$, for n > m, is there a square matrix $W\in R^{n\times n}$ so that $X^TWY$ is diagonal if and only if $Y = X$? Furthermore, $X$ and $Y$ are column normalized so that $X1_m = Y1_m = 1_m$, where $1_m$ is the m-length column vector with all entries equal to 1. </p> <p>I know that if $X$ and $Y$ are binary, then $W=I$.</p> http://mathoverflow.net/questions/92603/a-matrix-diagonalization-problem/92905#92905 Answer by silvanmx for A matrix diagonalization problem silvanmx 2012-04-02T15:51:41Z 2012-04-02T15:51:41Z <p>Here is a solution: $W = X(X^TX)^{-1}D(X^TX)^{-1}X^T$ for a diagonal matrix $D$. You may verify that $X^TWY = D$ iff $Y = X$.</p> http://mathoverflow.net/questions/21386/what-is-the-nonrecursive-formula-for-the-following-implicit-function What is the nonrecursive formula for the following implicit function? silvanmx 2010-04-14T20:42:05Z 2010-04-14T21:34:05Z <p>Given $f(0) = 1-c$, what is the non-recursive $f(n)$ that satify the following equation $f(n)-\frac{1}{2}f(n-1)f(n)+f(n-1) = 1$</p> <p>for n = 1,2,3,...?</p> http://mathoverflow.net/questions/19992/matrix-decomposition-problem Matrix decomposition problem silvanmx 2010-03-31T19:37:36Z 2010-04-06T14:18:30Z <p>Given a pair of distributions $x,y\in(0,1]^{n\times 1}$, so that $1^Tx=1$ and $1^Ty=1$, I want to build a matrix $C$ (change matrix) that satisfy <strong>at least</strong> the following properties:</p> <p>i) $C$ is diagonal if and only if $x=y$</p> <p>ii) $C1 = x$</p> <p>iii) $C^T1 = y$</p> <p>iv) $C$ has nonnegative entries</p> <p>How to build a $C$ that satisfy i)-iv)?</p> <p>If $\Lambda_x = diag(x)$ and $\Lambda_y = diag(y)$ conditions ii) and iii) can be also written as:</p> <p>(1) $C\Lambda_y^{-1}y = x$ </p> <p>(2) $C^T\Lambda_x^{-1}x = y$</p> <p>respectivelly. Replacing (2) in (1) results in:</p> <p>(3) $C\Lambda_y^{-1}C^T\Lambda_x^{-1}x = x$</p> <p>And replacing $x$ by $\Lambda_x1$ results the matricial Equation:</p> <p>(4) $(C\Lambda_y^{-1}C^T-\Lambda_x)1 = 0$</p> <p>or alternativelly (if 1 is replaced in 2), </p> <p>(5) $(C^T\Lambda_x^{-1}C-\Lambda_y)1 = 0$</p> http://mathoverflow.net/questions/96016/another-matrix-diagonalization-problem Comment by silvanmx silvanmx 2012-05-07T16:30:25Z 2012-05-07T16:30:25Z Regarding the properties of $Q$, consider this. Let $\Lambda_x = \mathrm{diag}(1_n^T X)$ and $\Lambda_y = \mathrm{diag}(1_n^T Y)$be the diagonal matrices formed with the row sum of $X$ and $Y$, respectively. Then, from ii) and iii), one can get $C 1_m = \Lambda_x 1_m$ and $1_m^TC = 1_m^T\Lambda_y$, for $C=Q^TX^TY^TQ$. These equations imply that the vector $1_m$ is a right eigenvector of $C$ with respect to $\Lambda_x$ and a left eigenvector with respecto to $\Lambda_y$. http://mathoverflow.net/questions/96016/another-matrix-diagonalization-problem Comment by silvanmx silvanmx 2012-05-07T16:27:45Z 2012-05-07T16:27:45Z BTW, could not figure out the typo. http://mathoverflow.net/questions/96016/another-matrix-diagonalization-problem Comment by silvanmx silvanmx 2012-05-07T16:00:21Z 2012-05-07T16:00:21Z Why is the problem interesting? Its solution will have relevance in land cover change detection. Each row of $X$ and $Y$ represents membership vectors of $m$ land cover types. If $X,Y\in\{1,0\}^{n\times m}$, i.e., are binary matrices with exactly one non-zero element per row, then $C=X^TY$ quantifies the changes among land cover types, and $C$ satisfies all three conditions above ($Q=I$). The problem arises when $X,Y\in[1,0]^{n\times m}$, that is partial memberships are allowed. http://mathoverflow.net/questions/92603/a-matrix-diagonalization-problem/92605#92605 Comment by silvanmx silvanmx 2012-05-04T19:22:53Z 2012-05-04T19:22:53Z I understand your point. However, for a propper solution of the problem one must be able to discern between diagonal and non-diagonal matrices. So, let's just discard the scalar case. – silvanmx 0 secs ago http://mathoverflow.net/questions/92603/a-matrix-diagonalization-problem/95587#95587 Comment by silvanmx silvanmx 2012-05-04T19:10:10Z 2012-05-04T19:10:10Z Looks like your answer is right! Thanks. http://mathoverflow.net/questions/92603/a-matrix-diagonalization-problem/92905#92905 Comment by silvanmx silvanmx 2012-05-04T19:08:05Z 2012-05-04T19:08:05Z You are right! That solution gives actually a specific one for a given $X$. It seems Spenser's reasoning above answers the question. http://mathoverflow.net/questions/92603/a-matrix-diagonalization-problem/92605#92605 Comment by silvanmx silvanmx 2012-04-02T16:31:12Z 2012-04-02T16:31:12Z Then what would be a non-diagonal 1-by-1 matrix? When I say ambiguous I mean that a scalar can be considered either diagonal or non-diagonal. http://mathoverflow.net/questions/92603/a-matrix-diagonalization-problem/92905#92905 Comment by silvanmx silvanmx 2012-04-02T16:16:35Z 2012-04-02T16:16:35Z If $Y\ne X$, then $X^TWX = D(X^TX)^{-1}X^TY$ which is (I'm guessing) non-diagonal. http://mathoverflow.net/questions/92603/a-matrix-diagonalization-problem Comment by silvanmx silvanmx 2012-04-02T15:54:44Z 2012-04-02T15:54:44Z I meant your second interpretation. Sorry for the confusion. http://mathoverflow.net/questions/92603/a-matrix-diagonalization-problem/92605#92605 Comment by silvanmx silvanmx 2012-03-29T20:15:27Z 2012-03-29T20:15:27Z Ok, I am assuming n&gt;&gt;m. Besides, the concept of a diagonal matrix is ambiguous for scalars. http://mathoverflow.net/questions/21386/what-is-the-nonrecursive-formula-for-the-following-implicit-function/21391#21391 Comment by silvanmx silvanmx 2010-04-15T21:12:04Z 2010-04-15T21:12:04Z What is CAS by the way? http://mathoverflow.net/questions/21386/what-is-the-nonrecursive-formula-for-the-following-implicit-function/21391#21391 Comment by silvanmx silvanmx 2010-04-15T21:11:35Z 2010-04-15T21:11:35Z Wow! Not tested yet but it looks like this is it. Thank you so much. http://mathoverflow.net/questions/19992/matrix-decomposition-problem/20433#20433 Comment by silvanmx silvanmx 2010-04-07T18:49:33Z 2010-04-07T18:49:33Z Will, your cost function works but I found it hard to justify. A maximal trace has more meaning for my application as it implies a minimum change, etc. So, the problem can be stated as: Maximize $tr(C)$ subject to $C1=1$, $C^T1=1$ and $C&gt;0$. I am just wondering if this problem accepts a unique solution or if there is an analytic solution out there. http://mathoverflow.net/questions/19992/matrix-decomposition-problem/20206#20206 Comment by silvanmx silvanmx 2010-04-06T15:09:51Z 2010-04-06T15:09:51Z Sorry, I forgot the nonnegativity constraint. The insigth is very useful though. http://mathoverflow.net/questions/19992/matrix-decomposition-problem/20433#20433 Comment by silvanmx silvanmx 2010-04-06T15:02:28Z 2010-04-06T15:02:28Z Interesting! It seems my problem now reduces to find an appropriate cost function.