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posted by  Harpo on 3/18/2008 10:46:41 AM  |  status: Closed  

linear algebra question

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I am trying to prove that for any square matrix A, A^T+A and AA^T is symmetric. I know there is a easier way to show this but I have drawn a blank. Can anyone help me? Any help is appreciated. Thanks
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Oracle
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posted by jks777 on 3/18/2008 11:00:03 AM  |  status: Live
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Harpo's comment:
"thank you so much for the help"
Response Details:
In order for a matrix to be symmetric, the matrix must be equal to its transpose (or A = AT).  In order to show that these matrices are symmetric, they must be equal to their transpose.
 
AT + A = (AT + A)T            Given (Definition of Symmetric Matrices)
AT + A = (AT)T + (A)T      Distributive Property of Transpose
AT + A = A + AT                  Transposes Cancel Out
AT + A = AT + A                  Communitive Property of Addition
 
Therefore AT + A is symmetrical.
 
AAT = (AAT)T                        Given (Definition of Symmetric Matrices)
AAT = (AT)TAT                      Transpose of Matrix Multiplication is Reversed
AAT = AAT                              Transposes Cancel Out
 
Therefore AAT is symmetrical.
 
 
 
Oracle
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posted by Ratso Rizzo on 3/18/2008 11:00:46 AM  |  status: Live
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Harpo's comment:
"thanks so much"
Response Details:
you need to show that aij = aji
Let (cij) = A^T + A
cij = aji + aij
cji = aij + aji
therefore, cij = cji, which implies symmetry.
Showing AA^T is symmetric is a bit tricker.
Later.
 
 
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posted by sandip on 3/18/2008 11:05:32 AM  |  status: Live
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Harpo's comment:
"thanks for taking the time to help me"
Response Details:
as we know that if A is symmetric matrix then it should obey
aij  =aji
which means A= AT----------1
AT+A=A+A= 2A which is a symmetric matrix [ since A is symmetric matrix]
A*AT=A*A  [from 1]
          =A2 which is again a symmetric matrix.
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