If $A$ and $B$ are symmetric matrices of same order then $AB+BA$ must be symmetric. But my question is what will happen for $AB-BA$. Is $AB-BA$ symmetric or skew-symmetric or is there no conclusion?
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I think it is skew-symmetric – Halima.Khatun Jan 07 '17 at 15:23
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Just check the definition of skew symmetry. What do you get? – user251257 Jan 07 '17 at 15:24
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1(AB-BA)^t=B^t A^t-A^t B^t=BA-AB=-(AB-BA) that implies it is skew-symmetric – Halima.Khatun Jan 07 '17 at 15:28
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yepp. Looks good. – user251257 Jan 07 '17 at 15:29
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Now I have another question transpose of symmetric matrix need not be symmetric matrix..plz explin me...clearly – Halima.Khatun Jan 07 '17 at 15:37
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again, check the definition. It is something you always try first. – user251257 Jan 07 '17 at 15:38
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a matrix A is symmetric if A^t=A.(A^t)^t=A=A^t..that implies transpose of symmetric matrix is symmetric – Halima.Khatun Jan 07 '17 at 15:43
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yes. looks good – user251257 Jan 07 '17 at 15:44