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Elementary matrix question

  1. Oct 22, 2008 #1
    I don't have a specific question in mind but can someone explain to me how to solve a question of the type " express the matrix A as the product of N elementary matrices"
     
  2. jcsd
  3. Oct 22, 2008 #2
    What you want to do is row reduce the matrix and keep track of your steps along the way. For example if you wanted to perform 3(Row1)+R3 to a matrix A, then you represent that by

    [tex]
    E=\begin{pmatrix}
    1&0&3 \\
    0&1&0 \\
    0&0&1
    \end{pmatrix}
    A[/tex]

    So make a matrix like that for each on of your steps; then if we call U your completely reduced matrix, then your original matrix, call it O is, [itex]O=LU[/itex] where [itex]L=E_1^{-1}E_2^{-1} \cdots E_n^{-1}[/itex]. Hope that helps.
     
  4. Oct 22, 2008 #3
    I almost get it other than how to keep track of the reduction steps... :) Can you explain to me how you formed that matrix that represents 3r1 - r3? I tried to figure it out but the best i can come up with is that the matrix you show me looks like the 3x3 identity matrix after the operation r1 + 3r3 ...
     
  5. Oct 22, 2008 #4
    Yes basically the way it works is you start with the identity matrix, and look at it vertically. For example,

    [tex]
    E=\begin{pmatrix}
    1&0&1 \\
    0&1&0 \\
    0&0&1
    \end{pmatrix}A[/tex]

    would do R1+R3, where

    [tex]
    E=\begin{pmatrix}
    1&0&0 \\
    0&1&0 \\
    2&0&1
    \end{pmatrix}A[/tex]

    would do 2R3+R1. Just insert the multiple of the row you want in that row and in the same column as the non-zero element of the row you want to add it to.
     
  6. Oct 22, 2008 #5
    ok i think i get it :)
    but how about operations whihc involve substracting rows (or does that count as scalar multiplication and addition in 2 separate steps) , and how about the operation of interchanging rows? how would that be represented
     
  7. Oct 22, 2008 #6
    If you want to interchange rows just use the identity matrix and switch the rows on there. For example interchanging row 2 and 3,

    [tex]
    E=\begin{pmatrix}
    1&0&0 \\
    0&0&1 \\
    0&1&0
    \end{pmatrix}A[/tex]

    and if you want to subtract a multiple of a row from another one just use a negative number as your multiple. For example, R3-3R1

    [tex]E=\begin{pmatrix}
    1&0&-3 \\
    0&1&0 \\
    0&0&1
    \end{pmatrix}A[/tex]
     
  8. Oct 22, 2008 #7
    ohhh excellent got it :) thank u very mcuh!!!!!
     
  9. Oct 22, 2008 #8
    no problem :]
     
  10. Oct 23, 2008 #9

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    In other words: the elementary matrix corresponding to a given row-operation is just the matrix you get by applying that row operation to the identity matrix.

    There are 3 kinds of row operations:
    1) swap two rows.
    2) multiply an entire row by a number
    3) add a multiple of one row to another.
     
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