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!

Finding limitations for matrix

  1. Feb 23, 2015 #1
    1. The problem statement, all variables and given/known data
    Let [itex]A\in Mat_{3,4}(K)[/itex]. Find all matrices X such that [itex]\forall X| A\cdot X = A'[/itex], where A' is the same as A with 2nd and 4th column swapped.

    2. Relevant equations

    3. The attempt at a solution
    First we determine the size of matrix X. By definition the first factor must have as many columns as the second has rows and the end product is [itex]A_{m,n}\cdot B_{n,p} = C_{m,p}[/itex]. X must be 4 x 4. Let [itex]a'_{i,j}\in A', a_{i,j}\in A[/itex]

    Haven't been able to (in my opinion) come up with anything sensible. Essentially, I have noticed that if we denote rows of A and columns of X as vectors, such that [itex]a_1 = (a_{1,1},a_{1,2},a_{1,3},a_{1,4})[/itex](by row up to a3) and [itex]x_1 = (x_{1,1},x_{2,1},x_{3,1},x_{4,1})[/itex](by column up to x4)
    Then the dot products equal to the corresponding element of A except when we multiply by either x2 or x4. Problem is, where do I find the conditions for matrix X?
    I can also write out the dot products individually, but then I have 3 equations and 4 variables..

    I have concluded that:
    a'_{i,j} = a_{i,j}= \sum_{k=1}^4 a_{i,k}\cdot x_{k,j},& j\in\{1,3\}\\
    a'_{i,2} = \sum_{k=1}^4 a_{i,k}\cdot x_{k, 2} = a_{i,4}\\
    a'_{i,4} = \sum_{k=1}^4 a_{i,k}\cdot x_{k, 4} = a_{i,2}


    THIS LOOKS UGLY! I can't think of any other way to show conditions for all x in X :s
    This definitely is not acceptable, suggestions?

    (I want to avoid writing out individual x - what if the dimensions of the matrix are countible? Would take a long time to write them out xD)
  2. jcsd
  3. Feb 23, 2015 #2


    User Avatar
    Science Advisor

    If X were the identity matrix it would look like this: [itex]
    1 & 0 & 0 & 0 \\
    0 & 1 & 0 & 0\\
    0 & 0 & 1 & 0 \\
    0 & 0 & 0 & 1 \\
    \end{matrix}[/itex]. The second row controls the second column in the product, and the fourth row controls the fourth column.
  4. Feb 23, 2015 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Google 'matrix column operations'; for example, see

    These show explicitly how to find the matrix X. (They leave unanswered the question of whether X is unique---they just show how to find one possible X.)
  5. Feb 24, 2015 #4
    Okay, if I approach this deductively, then
    [itex]a'_{11} = a_{11}\cdot x_{11} + a_{12}\cdot x_{21} + a_{13}\cdot x_{31} + a_{14}\cdot x_{41} = a_{11} [/itex] if x11 = 1, then the other three summands would sum to 0 and the resulting matrix is very similar to the identity matrix, BUT
    x11 does not have to be 1 in which case everything breaks.

    Intuitively I am quite sure that the X is the identity matrix with 2, 4 column swapped - but this is all deduction based math. I am not satisfied.
  6. Feb 24, 2015 #5

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    What is unsatisfactory about it? What is wrong with building on past knowledge developed by others?
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: Finding limitations for matrix
  1. Finding a limit (Replies: 1)

  2. Find this limit (Replies: 12)

  3. Finding Matrix A (Replies: 1)

  4. Find matrixes B and C? (Replies: 2)