we have to show that every square matrix can be written as the sum of two commuting matrices...i dont know how to do it in general.i think we have to prove if A=B+C then BC=CB...am i ri8....what will b further approach about this question?but B and C should not be the multiple of identity matrix or zero matrix.
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are you sure this is true? – Feb 22 '14 at 13:55
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yes this is true.... – hafsah Feb 22 '14 at 13:56
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forget about its correctness but according to the question you are asked to prove that there exists matrices $B,C$ such that $A=B+C$ and $BC=CB$ and not that if $A=B+C$ then $BC=CB$ – Feb 22 '14 at 13:58
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what will be apropriate method to prove it?? @PraphullaKoushik – hafsah Feb 22 '14 at 14:00
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This would be an extremely boring thing to prove, since $A=A+0$ and $A\cdot0=0\cdot A$. – Carsten S Feb 22 '14 at 14:04
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if we are not allowed to take zero matrix..then....@CarstenSchultz?? – hafsah Feb 22 '14 at 14:07
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I guess this should help : $$\begin{bmatrix}a&b\\c&d\end{bmatrix}=\begin{bmatrix}a-1&b\\c&d-1\end{bmatrix}+\begin{bmatrix}1&0\\0&1\end{bmatrix}$$
Can you prove that the two matrices in the right side would commute?
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yes ofcource ....but isnt there any choice other then identity matrix?? – hafsah Feb 22 '14 at 14:07
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No idea as of now.. would inform you if i come across such.... you might want to mention that in the question... "I need two matrices $B$ and $C$ neither of them being multiples of identity matrices" – Feb 22 '14 at 14:13
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