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Basic Linear Algebra Problem

  1. Sep 25, 2009 #1
    1. The problem statement, all variables and given/known data

    Let L and U be invertible 3x3 matrices, L is lower triangular and U is upper triangular. Assume LU is upper triangular, show that L is a diagonal matrix.

    2. Relevant equations

    None? Maybe A=LU or A=LDU

    3. The attempt at a solution

    Basically, I can reason out the beginnings of a solution but I can't get it to be concrete enough. I am thinking that if we let LU=A, A is upper triangular, then U and A are of the same form. Thus we could think of A=LU as the standard factorization of A using Elimination (Elementary Row Matrices) Matrices to get A into upper triangular to presumably solve a system of three equations. But since A is already in the desired form, there are no row operations of eliminations to be made, this all the multipliers are exactly zero this every term of the lower triangular inverse elimination matrix below the main diagonal (where one would place the multipliers) is zero, this L is a diagonal matrix.

    This isn't necessarily a proof based class, and this question isn't asking for a rigorous proof, but I still feel like the answer above is weak and would like to strengthen it. Or, if it is completely wrong in reasoning, then obviously I would like to arrive at the correct method of reasoning. Any suggestions? Thanks in advance.
     
  2. jcsd
  3. Sep 25, 2009 #2
    Can you write down a formula for the i,j component of A in terms of the components for U and L?
     
  4. Sep 26, 2009 #3
    You can use the splitting of a matrix in the basis Ei,j(=1 if i=j, 0 otherwise).
    L and U are inversible implies their diagonal components are different from 0.
    And use the formula that expresses LxU.
     
  5. Sep 26, 2009 #4
    Thanks guys, I solved this shortly after posting. To anyone else who might come across this, the above posts are what to do, also remember the matrices are invertible and no diagonal entries can be zero...
     
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