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Find Basis for diagonal matrix

  1. Oct 27, 2012 #1
    I'm not sure how to start this problem.
    All i know is a diagonal matrix consists of all 0 elements except along the main diagonal.

    But how do I even find a basis for this?
     

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  3. Oct 27, 2012 #2

    haruspex

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    What would a basis look like? It would be set of nxn matrices such that... you can do what with them?
     
  4. Oct 27, 2012 #3
    For this case, a basis consists of all matrices such that all nxn diagonal matrices can be written as a linear combination of them?
     
  5. Oct 27, 2012 #4

    haruspex

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    Yes. What's the simplest matrix you can think of that might be useful in creating such a basis?
     
  6. Oct 27, 2012 #5
    This is where I get stuck. I've only been taught and done problems where the basis is a set of "vectors."

    I saw somewhere that the basis for a 2x2matrix is
    1 0
    0 0

    0 1
    0 0

    0 0
    1 0

    0 0
    0 1


    if it were a 2x2 diagonal would it be
    1 0
    0 0

    and
    0 0
    0 1
    ?
     
  7. Oct 27, 2012 #6

    haruspex

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    Ok. Now try 3x3.
     
  8. Oct 27, 2012 #7
    1 0 0
    0 0 0
    0 0 0

    0 0 0
    0 1 0
    0 0 0

    0 0 0
    0 0 0
    0 0 1


    if it's an nxn matrix, wouldn't that give an infinite amount of matrices for the bases?


    The answer in the back of the book is
    37. B = {eii | 1 ≤ i ≤ n} the "ii" part is supposed to be subscripts for e. I'm bad at interpreting these kind of answers, what is it saying?
     
  9. Oct 27, 2012 #8

    haruspex

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    You had 2 for 2x2 and 3 for 3x3. Why would you get infinitely many for nxn?
     
  10. Oct 27, 2012 #9
    oops i mean n amount.
     
  11. Oct 27, 2012 #10

    haruspex

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    So you have the answer.
    The eii notation used in the book apparently means the nxn matrix that has 1 at the (i, i) position and 0 everywhere else. I don't know how standard that is. Should be defined in the book somewhere.
     
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