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-   -   Symmetric matrix (https://www.askmehelpdesk.com/showthread.php?t=177445)

  • Jan 27, 2008, 09:24 PM
    cool_dude
    symmetric matrix
    The question I have is:

    Let A be an n x n symmetric matrix. Show that A^k is symmetric if k is any nonnegative integer.

    I know this is true but I'm a little confused in how to show it. I can take any symmetric matrix and then take any exponent n and see that its symmetric. But is there any other way to show this other than by example?

    Thanks
  • Jan 28, 2008, 05:55 PM
    galactus
    Cool problem. I think we can prove this with induction on k.

    If A is symmetric, then so is when k=1.

    (Also, since , the result is true for k=0.)

    Let's assume that is symmetric for k=1,. n for some .

    Then, is the product of two symmmetric matrices. So,

    by the properties of the transpose :



    And therefore, hence and heretofore is also symmetric.

    Thus, is symmetric for all .
  • Jan 28, 2008, 09:01 PM
    cool_dude
    Neat solution. I was thinking somewhere along the lines of induction but ended up prooving it by just using some of the properties for symetric and transpose matrices. I handed it in so when I get it back I'll see if this was the correct approach. Thank you.

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