Need Help: New kind of Matrix formula?

Replies

• 

  ErAnushka

  No problem I got 😀
  
  A direct proof is as follows:
  
  Let X be the given square matrix
  
  Let X=A+B where A is symmetric and B is skew symmetric
  
  That is A'=A and B' = - B .......(1)
  Therefore,
  X'=A'+B' ( using the property of transpose of a matrix)
  or, X'=A - B from (1) above
  
  Now, as X is square , therefore, X+X' and X-X' are defined.
  
  Therefore X+X'=A+B +A-B= 2A
  
  and X-X'=A+B- (A-B) = 2B
  
  Therefrore A=(X+X')/2
  
  and B= (X-X')/2
  
  It is easy to verify that A'=A and B'= - B
  
  That is A is symmetric and B is skew symmetric
  
  Therefore , X can be uniquely expressed as sum of a symmetric matrix and a skew symmetric matrix, which is
  
  X =(X+X')/2 + (X-X')/2
  
  
  
  
  I understood the green part 😀 , but the red part is little confusing😕 .. A'=A and B' = -B but how? we got only A=(X+X')/2 and B= (X-X')/2.... ??? 😨
• 

  silverscorpion

  I'm not sure what you are not getting here.. A = A' and B = -B' are assumptions that you made in the beginning, right? (ie., you have assumed that A is symmetric and B is skew symmetric)
  
  You can probably take the transpose of (X+X')/2 and (X-X')/2, and check that they are indeed the same as A and -B.. but, here it's actually assumed that it is true.
• 

  ErAnushka

  silverscorpion

  I'm not sure what you are not getting here.. A = A' and B = -B' are assumptions that you made in the beginning, right? (ie., you have assumed that A is symmetric and B is skew symmetric)
  
  You can probably take the transpose of (X+X')/2 and (X-X')/2, and check that they are indeed the same as A and -B.. but, here it's actually assumed that it is true.

  Omg wait... I got it 😀
  
  We are actually proving that any square matrix is a sum of symmetric matrix and skew symmetric matrix, so here, A is symmetric matrix and B is skew symmetric matrix.
  
  We got both value, so,
  
  A+B = (X+X')/2 + (X-X')/2
  A+B = (X+X' + X-X') /2
  A+B = 2X/2
  A+B = X
  
  Here X is a square matrix! OMG! I'm so stupid 😁
  By the by thanks for the reply, I got the idea at the moment of reading this again 😀