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Quick linear algebra determinant proof.

  1. Jan 28, 2012 #1
    1. The problem statement, all variables and given/known data

    If A is a square symmetric matrix nxn. Show that the determinant of A is the product of its eigenvalues.

    2. Relevant equations



    3. The attempt at a solution

    From spectral decomp.

    A = QλQ'
    |A| = |QλQ'| = |QQ'λ| = |Q||Q'||λ| = |λ| = the product of its diagonals (eigenvalues).

    The step I'm not 100% sure of is if I can interchange QλQ' to QQ'λ
     
  2. jcsd
  3. Jan 28, 2012 #2

    I like Serena

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    Homework Helper

    Hi Kuma! :smile:

    No, you can't interchange QλQ' to QQ'λ.
    This can be seen because then QQ'λ=Iλ=λ, but this would not match your original matrix A.

    However, there is no need to interchange them.
    You can take the same steps without interchanging them.
     
  4. Jan 28, 2012 #3

    Ray Vickson

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    The constant in the characteristic equation of A equals the determinant of A. How is that constant related to the eigenvalues?

    RGV
     
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