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September 29th, 2008, 01:35 AM   #1
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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)

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September 29th, 2008, 02:34 AM   #2
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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.
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January 26th, 2009, 11:12 AM   #3
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Re: Matrix operations proof problem.. Help please..

Let X=(A+A^T)/2 , Y=(A-A^T)/2
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