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anonymous

  • one year ago

let A be an m*n matrix. show that A^T *A and A*A^T are both symetric ??

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  1. amoodarya
    • one year ago
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    how can m*n matrix be symmetric ?

  2. anonymous
    • one year ago
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    A^T*A is symetric not A

  3. amoodarya
    • one year ago
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    \[A_{m*n} \rightarrow (A_{m*n} )^T=A'_{n*m} \] transpose of m*n matrix is a n*m matrix ,and when \[m \neq n\] \[A^T \neq A\]

  4. anonymous
    • one year ago
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    A^T isnt symmetric but A*A^T is symmetric read it again !

  5. amoodarya
    • one year ago
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    sorry but I think , the question changed !?

  6. amoodarya
    • one year ago
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    it suffice to apply transpose , to check symmetry \[(A^T*A)^T=\\(A^T)*(A^T)^T=\\A^T*A\]

  7. amoodarya
    • one year ago
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    \[(A*A^T)^T=\\(A^T)^T*A^T=\\A*A^T\]

  8. anonymous
    • one year ago
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    thanx

  9. imqwerty
    • one year ago
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    a useful link - http://users.math.msu.edu/users/hhu/309/3091326.pdf

  10. amoodarya
    • one year ago
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    \[(AA^T)_{m*m}=(A_{m*n} \times A^T_{n*m})^T=\\(A^T_{n*m})^T \times (A_{m*n})^T=\\(A_{m*n})\times A^T_{n*m}=\\(AA^T)_{m*m}\]

  11. Empty
    • one year ago
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    I don't understand these subscripts being introduced, it seems sorta extra information, we already know that A is an mxn matrix so it doesn't really tell us anything we wouldn't already know. Really this whole thing just hinges on the fact of how the transpose has this property: \[(AB)^T = B^TA^T\] So if we want to prove something is symmetric we just have to show: \[M=M^T\] so we take the matrix: \[(AA^T)^T=A^{TT}A^T=AA^T\] And we're done once we know transposing twice is the same as not having transposed at all.

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