anonymous
  • anonymous
let A be an m*n matrix. show that A^T *A and A*A^T are both symetric ??
Linear Algebra
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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amoodarya
  • amoodarya
how can m*n matrix be symmetric ?
anonymous
  • anonymous
A^T*A is symetric not A
amoodarya
  • amoodarya
\[A_{m*n} \rightarrow (A_{m*n} )^T=A'_{n*m} \] transpose of m*n matrix is a n*m matrix ,and when \[m \neq n\] \[A^T \neq A\]

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anonymous
  • anonymous
A^T isnt symmetric but A*A^T is symmetric read it again !
amoodarya
  • amoodarya
sorry but I think , the question changed !?
amoodarya
  • amoodarya
it suffice to apply transpose , to check symmetry \[(A^T*A)^T=\\(A^T)*(A^T)^T=\\A^T*A\]
amoodarya
  • amoodarya
\[(A*A^T)^T=\\(A^T)^T*A^T=\\A*A^T\]
anonymous
  • anonymous
thanx
imqwerty
  • imqwerty
a useful link - http://users.math.msu.edu/users/hhu/309/3091326.pdf
amoodarya
  • amoodarya
\[(AA^T)_{m*m}=(A_{m*n} \times A^T_{n*m})^T=\\(A^T_{n*m})^T \times (A_{m*n})^T=\\(A_{m*n})\times A^T_{n*m}=\\(AA^T)_{m*m}\]
Empty
  • Empty
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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