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Re: Re: wrong answer or no answer?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg82867] Re: [mg82812] Re: wrong answer or no answer?
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Thu, 1 Nov 2007 05:25:00 -0500 (EST)
*References*: <200710291054.FAA06915@smc.vnet.net> <fg6pf3$ckv$1@smc.vnet.net> <200710311119.GAA22682@smc.vnet.net>
On 31 Oct 2007, at 20:19, Roger Bagula wrote:
> Andrzej Kozlowski wrote:
>
>> Your matrix M has the property:
>>
>> MatrixPower[M, 2]
>> {{0, 0}, {0, 0}}
>>
>> It is easy to prove using elementary linear algebra that such a
>> matrix has no square root. In fact one can prove more. Suppose than M
>> is an n by n matrix such that M^n=0 but M^(n-1) !=0 (in other words M
>> is nilpotent of order n). Then M has no square root.
>>
>> The proof is easy so I won't bother to give it here.
>>
>> Andrzej Kozlowski
>>
>>
>>
>>
> Andrzej Kozlowski
> It certainly does have this property.
> Your second post is probably best :
>
> If Det[M]=0, then MatrixPower[M,n]=0 not matter what n is.
>
This is completely false (and I did not write anything of the kind).
M = {{0, 0, 0}, {1, 0, 0}, {1, 1, 0}}
Det[M]
0
but
MatrixPower[M, 2]
{{0, 0, 0}, {0, 0, 0}, {1, 0, 0}}
That is not the zero matrix.
Andrzej Kozlowski
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