User Name Remember Me? Password

 Linear Algebra Linear Algebra Math Forum

 November 17th, 2015, 10:26 AM #1 Newbie   Joined: Oct 2015 From: CR Posts: 10 Thanks: 0 Regular matrix - LU decomposition I have the following math problem: Determime for which ${x,y,z}$ is the matrix $A$ regular and then find the LU decomposition. $A=\begin{pmatrix} 1 & 1 & 1 \\ x & y & z \\ x^2 & y^2 &z^2\\ \end{pmatrix}$ Determining when it is regular... I have tried using the definition of regular matrix. For instance that its rows and columns must be linearly independent but that didn´t take me far. We know the rank is 3 but we canť really use that anywhere.. From the definition we know that when a matrix is regular it is also invertible. That implies there exists some matrix $X$ (inverse of $A$) such that $AX=XA=I_n$.. In this particular case $I_3$ Knowing that, if I now use the definition of matrix multiplication we get 9 equations with 12 variables? That doesn´t do anything for me either, does it? I am still not sure how to proceed to find the general solution (all triplets of x, y, z). I would also appreciate an approach without using determinant. Thank you. November 18th, 2015, 06:36 AM #2 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 Det A=(x-y)(x-z)(y-z) which is non-singular if x, y, and z are unequal. A is called a Van der Monde determinant. For the LU decomposition (a row reduction procedure) google"LU decomposition" and watch one of the utubes. November 18th, 2015, 10:41 AM #3 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 $\displaystyle \begin{vmatrix} 1 &1 &1 \\ x&y &z \\ x^{2}&y^{2} &z^{2} \end{vmatrix}=\begin{vmatrix} 1 & 1 & 1\\ x& y &z \\ 0 &y^{2}-xy & z^{2}-xz \end{vmatrix}=\begin{vmatrix} 1 &1 & 1\\ 0 &y-x &z-x \\ 0 & (y-x)y & (z-x)z \end{vmatrix}=\\(y-x)(z-x)\begin{vmatrix} 1 &1 \\ y & z \end{vmatrix}=(z-x)(z-y)(y-x)$ You didn't see that? I had no problem once I saw someone else do it. By the way, that's basically how you prove the general case by induction. Tags decomposition, matrix, regular Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post pej Linear Algebra 0 October 13th, 2015 07:43 PM mahjk17 Linear Algebra 4 September 10th, 2012 01:29 PM Muthuraj R Linear Algebra 2 February 11th, 2012 03:45 PM Sergio Economics 0 June 6th, 2010 02:03 AM enrigue Linear Algebra 1 January 26th, 2009 07:15 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top       