1. Not finding help here? Sign up for a free 30min 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!

Prove that a matrix can be reduced to RRE and CRE

  1. Jun 9, 2017 #1
    1. The problem statement, all variables and given/known data

    Let ##A## be an ##m \times n## matrix. Show that by means of a finite number of elementary row/column operations ##A## can be reduced to both "row reduced echelon" and "column reduced echelon" matrix ##R##. i.e ##R_{ij} = 0## if ##i \ne j##, ##R_{ii} = 1 ##, ##1 \le i \le r##, ##R_{ii} = 0## if ##i > r##. Also show that ##R = PAQ## where ##P## and ##Q## are invertible ##m\times m## and ##n \times n## matrices respectively.


    2. Relevant equations


    3. The attempt at a solution

    Since I know I can pass ##A## to a row reduced echelon matrix in finite number of operations.
    Lets say the row reduced echelon form of ##A## is ##R^{\prime}##. Then ##R^\prime = PA##.

    Also since nothing is special about rows, therefore I can say that a matrix can be passed on to column reduced echelon in finite number of steps. Therefore I can pass ##R^\prime## to a column reduced form ##R## in finite number of steps. Lets say ##R = QR^\prime##

    From above I can say ##A## can be passed to a column and row reduced echelon form in finite number of steps and ##R = QPA##.

    Is this correct ? I think it is wrong since I used a lot of words and also I got ##R = QPA## not ##R = PAQ##.
     
  2. jcsd
  3. Jun 9, 2017 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    So, you can have ##R = QPA,## and this can be written as ##R = P_1 A Q_1##, where ##P_1 = QP## and ##Q_1 = I## (the ##n \times n## identity matrix).
     
  4. Jun 9, 2017 #3
    Then the proof is correct ?
     
  5. Jun 9, 2017 #4

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    What proof? All you did was make statements; you did not really "prove" anything.
     
  6. Jun 9, 2017 #5
    No I did not get what you are saying. Isn't statements like " You can pass from from A to a row/column reduced form in finite steps " is proven to be true.
    So I just need to combine these types of statements to form a proof.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted