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Equivalent matrix?

  1. May 6, 2007 #1
    1. The problem statement, all variables and given/known data
    How do you turn the transformation matrix

    1+i 3
    2-i 5i

    into a diagonal matrix when it is transforming vectors in the Z module?

    What is the Z module? Normally one states R module M. where R is the coefficient one multiplies to the elements in the module.

    3. The attempt at a solution
    I have tried but can't seem to reduce to diagonal form as I can only multiply by integer numbers?
  2. jcsd
  3. May 6, 2007 #2
    I have just worked out the diagonal matrix. It is

    1 0
    0 8+11i

    However my question of what is the Z module still stands. I know clearly what a Z ring is.
  4. May 6, 2007 #3


    User Avatar
    Staff Emeritus
    Science Advisor

    Z is the same whether you think of it as a ring or a module- the "Gaussian integers"- all numbers of the form a+ bi where a and b are integers and i is, of course, the imaginary unit.
  5. May 6, 2007 #4
    I think that the Z module is refering to any module where the coefficients are from the ring of the Gaussian integers. Why use the word Gaussian?
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