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Mathematics of circular shifts of rows and columns of a matrix

  1. Mar 20, 2014 #1
    What mathematical algebra explains sequence of circular shifts on rows and columns of a matrix?

    Consider a simple matrix (3X3) with entries thus: [1 2 3; 4 5 6; 7 8 9;]

    Circular shifts can be performed on any row or any column thus: row-(1/2/3)-(right/left) and column-(1/2/3)-(up/dn)

    Examples: R1-right transforms [1 2 3] to [3 1 2]. R3-left makes [7 8 9], [8 9 7]. Also: C1-up converts [1 4 7] to [4 7 1].

    Now these moves can be performed repeatedly on the initial matrix. Assume a sequence of moves thus: R1-r, C1-up, R3-left, C2-dn, R2-r, C3-up

    This converts the initial matrix to this matrix (you can work this to confirm): [4 9 1; 6 7 3; 8 5 2].

    Now assume that you know NONE of these moves?

    Given simply the 2 matrices: the initial and the final, what method shall help me find the moves that lead from the initial to final, or final to initial. The latter shall be an inverted sequence of each former moves' reverse.

    What algebra goes here in? Please point to any and every resource like group theory, number theory, permutation theory, sequential circuits etc...

    This matrix 'jumbling' if you say so, inspired in part from the Rubik's Cube finds some interesting applications in cryptography and data transformation.

    I look forward to your comments.
     
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  3. Mar 20, 2014 #2

    HallsofIvy

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    Your first problem is that as soon as you have more than one "shift", there will be more than one set of shifts that will give the same result. So there cannot be one unique answer.
     
  4. Mar 20, 2014 #3

    Stephen Tashi

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    Is a circular shift "on a row" of the matrix to be an operation that only does a circular shift on the elements of that row? Or does it perform that operation on each row of the matrix?

    A matrix with exactly a single 1 in each row and in each column and the rest of the entries 0 represents a permutation. if we think of , say, the column vector C being multiplied on the left by such a matrix, the resulting product is column vector that is a permutation of the elements of C. If you thought of the elements of MxN matrix written as one long column vector with Mx N components, you could find such a (MxN) by (MxN) "permutation matrix" that did a circular shift on the elements that compose a single row. From that point of view, the relevant algebra is group theory. (The relevant advanced topic is "group representations".)
     
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