
Linear Algebra Linear Algebra Math Forum 
 LinkBack  Thread Tools  Display Modes 
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: 125 
Det A=(xy)(xz)(yz) which is nonsingular 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: 125 
$\displaystyle \begin{vmatrix} 1 &1 &1 \\ x&y &z \\ x^{2}&y^{2} &z^{2} \end{vmatrix}=\begin{vmatrix} 1 & 1 & 1\\ x& y &z \\ 0 &y^{2}xy & z^{2}xz \end{vmatrix}=\begin{vmatrix} 1 &1 & 1\\ 0 &yx &zx \\ 0 & (yx)y & (zx)z \end{vmatrix}=\\(yx)(zx)\begin{vmatrix} 1 &1 \\ y & z \end{vmatrix}=(zx)(zy)(yx)$ 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  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
On Matrix Decomposition  pej  Linear Algebra  0  October 13th, 2015 07:43 PM 
A regular matrix problem  mahjk17  Linear Algebra  4  September 10th, 2012 01:29 PM 
Matrix decomposition  Muthuraj R  Linear Algebra  2  February 11th, 2012 03:45 PM 
Urgent Eigenvector decomposition of a matrix  Sergio  Economics  0  June 6th, 2010 02:03 AM 
Polar Decomposition of a nxn matrix  enrigue  Linear Algebra  1  January 26th, 2009 07:15 AM 