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Normal Matrices Examples ( )

  1. Nov 29, 2012 #1
    Normal Matrices Examples (URGENT)

    I need to produce 2 x 2 normal matrices A and B such that A + B is not normal. I have proven that AB is normal if AB = BA using the Householder matrix form. But I can't find a form for A + B failing to be normal.

    A matrix A with entries a, b, c, d is going to be normal if b = c. But that means a matrix B has to have b = c to be normal as well. So A + B will have 2b, 2c for the 1,2 and 2,1 entries as well which will make A + B normal.

    Unless I'm totally off. I have to get this done in the next 3 hours.
     
  2. jcsd
  3. Nov 29, 2012 #2

    micromass

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    Re: Normal Matrices Examples (URGENT)

    Please don't write things like

    or "urgent". You will actually get less replies that way.
     
  4. Nov 29, 2012 #3

    berkeman

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    Re: Normal Matrices Examples (URGENT)

    Schoolwork?
     
  5. Nov 29, 2012 #4

    mathwonk

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    Re: Normal Matrices Examples (URGENT)

    relax, the world will not end if this drags out.

    I suggest starting by thinking up a really simple non normal matrix, then try to realize it as A+B where A and B are a little nicer. (This took me about 30 seconds.)
     
  6. Nov 29, 2012 #5

    micromass

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    Re: Normal Matrices Examples (URGENT)

    I've moved it to homework.
     
  7. Nov 29, 2012 #6
    Re: Normal Matrices Examples (URGENT)

    I don't understand how to make a non-normal matrix break down into 2 normal matrices. That's the whole point of my post.
     
  8. Nov 29, 2012 #7

    micromass

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    Re: Normal Matrices Examples (URGENT)

    Can you give us an example of a non-normal matrix?
     
  9. Nov 29, 2012 #8
    Re: Normal Matrices Examples (URGENT)

    1 1
    0 1

    There is no way to make that matrix the sum of 2 normal matrices A and B?
     
    Last edited: Nov 29, 2012
  10. Nov 29, 2012 #9

    Dick

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    Re: Normal Matrices Examples (URGENT)

    Sure there's a way. Try and write it as the sum of a symmetric matrix and an antisymmetric matrix.
     
    Last edited: Nov 29, 2012
  11. Nov 29, 2012 #10
    Re: Normal Matrices Examples (URGENT)

    Now I understand. I used

    0 1
    1 0

    and

    0 -1
    1 0
     
  12. Nov 29, 2012 #11

    Dick

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    Re: Normal Matrices Examples (URGENT)

    Sure. That works.
     
  13. Nov 29, 2012 #12
    Re: Normal Matrices Examples (URGENT)

    Love being in grad school and missing stupid things like that. -__- Preciate the help.
     
  14. Nov 29, 2012 #13

    mathwonk

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    Re: Normal Matrices Examples (URGENT)

    nice hint, micro mass. good job, tazzzdo.
     
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