1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Triangular Matrices

  1. Jan 12, 2015 #1
    1. The problem statement, all variables and given/known data
    Prove that is ##A## is lower triangular and ##B_{ij}## is the matrix that results when the ith row and jth column of A are deleted, then ##B_{ij}## is lower triangular if i > j.

    2. Relevant equations


    3. The attempt at a solution

    I know that a square matrix is lower triangular if and only if the jth column starts with at least j-1 zero's for every j.

    I am attemting to prove this by contradiction.
    If i = j, then the jth column of B has j-1 zeros, but this is true?
    if i was to take a 4 x 4 lower triangluar matrix and delete its first row and first column would the 1st,2nd,3rd,4th columns have 0,1,2,3 zeros respecively?


    Lets ignore that, prehaps im saying/doing something silly, to finish my proof by contradiction i also have to show that for i < j, we dont have a lower triangular matrix.

    if the ith row and the (i+n)th collumn, where n is a postitive integer, of A is deleted, then the ...

    ah im really not getting this one guys. im lost.
     
  2. jcsd
  3. Jan 12, 2015 #2
    Have i not followed the forums rules? if not could someone please point out the error of my ways
     
  4. Jan 12, 2015 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    I think you have not done "enough" work on your own to satisfy PF requirements. You ask: "if i was to take a 4 x 4 lower triangluar matrix and delete its first row and first column would the 1st,2nd,3rd,4th columns have 0,1,2,3 zeros respecively?" I do not see why you cannot answer this for yourself: just draw a sketch of what a 4 x 4 lower triangular matrix must look like, then remove its first row and column. What does it look like now? Try to generalize this to removing row ##r## and column ##c## from a lower-triangular matrix, where ##r = c##. What does the new matrix look like? Ditto if ##r > c##.
     
  5. Jan 14, 2015 #4
    Thanks, Ray. Maybe i should have made what i HAD done already a little more clear.

    I have already done what you said, regarding a 4x4 matrix and eliminating the first row and column, what results is another lower triangular matrix. the question asks me to show that that if i > j then ##B_{ij}## is a lower triangluar matrix. but in the 4 x 4 example, when i = j, ##B_{ij}## is a lower triangular still, so is proof by contradtiction not the way to go?

    in summary;
    I have found by experiment, when i = j, ##B_{ij}## is still lower triangular
    Also using a 4 x4 matrix as an experiment, when the first row and second column are deleted, what results is another lower triangular matrix. that is i < j

    finally, when the second row and the first column are deleted, that is i > j, now, this is not lower triangular, since the first row is a row of zeros.

    Prehaps im not understanding the question, but what is evedent from experiment is that the opposite of what i have to prove is true.
    I hope my definition of a lower triangular is correct;
    that is, each entry to the right of the main diagonal is zero.

    That is where my confusion lies.
     
    Last edited: Jan 14, 2015
  6. Jan 15, 2015 #5

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    When you omit row 2 and column 1 you get a matrix that is again lower triangular. The fact that the first row is all zero is irrelevent; all that matters is that elements strictly to the right of the diagonal are zero---and they are in your case.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Triangular Matrices
  1. Triangular Prism (Replies: 4)

  2. The Matrices (Replies: 12)

Loading...