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Symmetric matrices

  1. Dec 11, 2008 #1
    Give an example of a 2X2 symmetric matrix B that cannot be written as B = ATA. Give an explanation as to why no such A exists for the matrix B you have given.


    I know that the product ATA is a symmetric matrix, but how could there be no such A that exists for some matrix B?

    I'm really stuck on this problem, and I would appreciate it if anyone could help. Thank you so much in advance.
     
    Last edited: Dec 11, 2008
  2. jcsd
  3. Dec 11, 2008 #2

    D H

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    What is the nature of the eigenvalues of ATA?
     
  4. Dec 11, 2008 #3

    Fredrik

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    If you write

    [tex]A=\begin{pmatrix}a&b\\ c&d\end{pmatrix}[/tex]

    and calculate ATA, it's immediately obvious that the result is not the most general form of a symmetric 2×2 matrix. No more calculations are necessary. Just stare at the result until you get it.
     
  5. Dec 11, 2008 #4
    1 2
    2 3

    I did out the product of ATA and came up with this matrix B that couldn't be written as such. Am I correct?
     
  6. Dec 11, 2008 #5
    You are correct about the above matrix. Try to figure out why (hint: determinants).
    What do you think about the following matrix:
    [tex]
    A=\begin{pmatrix}1&10\\ 10&-1\end{pmatrix}
    [/tex]
     
  7. Dec 12, 2008 #6

    D H

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    Determinants are quite the right answer, either. Both the identity matrix and its additive inverse,

    [tex]\bmatrix -1 & 0 \\ 0 & -1\endbmatrix[/tex]

    have the same determinant. The identity matrix can obviously be written in the form ATA. Its additive inverse cannot.
     
  8. Dec 12, 2008 #7

    Fredrik

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    The OP's task was just to find an example of a symmetric matrix that can't be expressed as ATA, and noting that det ATA≥0 is certainly a good start. It explains why

    1 2
    2 3

    is the kind of matrix we're looking for. My idea was to note that the elements on the diagonal of ATA are ≥0. That explains both

    1 10
    10 -1

    and

    -1 0
    0 -1

    Not sure if there are symmetric matrices with both the determinant and the diagonal elements ≥0 that cant be written as ATA. I'm too tired to think about that right now.
     
  9. Dec 12, 2008 #8

    D H

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    You can get away with looking at the determinant and the diagonal elements for a 2x2. That trick won't work for anything larger than 2x2. What always works is to look at the eigenvalues, like I said in post #2. Given a real nxm matrix A, the eigenvalues of the matrix ATA are always non-negative. If any eigenvalue of some matrix B is negative then it is impossible to write B in the form ATA.
     
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