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September 29th, 2008, 01:35 AM  #1 
Member Joined: Dec 2007 Posts: 30 Thanks: 0  Matrix operations proof problem.. Help please..
I have some trouble with the following proof. Any help would be great! Let A be an n x n matrix. Show that A can be uniquely written A= X+Y, where X is symmetric (that is X^T=X) and Y is antisymmetric (that is, Y^T=Y). [The usage Y^T represents the transpose of matrix Y) Thanks! 
September 29th, 2008, 02:34 AM  #2 
Site Founder Joined: Nov 2006 From: France Posts: 824 Thanks: 7  Re: Matrix operations proof problem.. Help please..
Consider two coefficients a_ij and a_ji which are symmetric with respect to the principal diagonal of your matrix A=(a_mn). Your problem is equivalent with showing that there exists two numbers x_ij and y_ij such that: a_ij = x_ij + y_ij and a_ji = x_ji + y_ji = x_ij y_ij. This system of equations has determinant 2 and thus admits a unique solution. This solves your problem.

January 26th, 2009, 11:12 AM  #3 
Member Joined: Jan 2009 Posts: 72 Thanks: 0  Re: Matrix operations proof problem.. Help please..
Let X=(A+A^T)/2 , Y=(AA^T)/2


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