Re: Questions regarding MatrixExp, and its usage

• To: mathgroup at smc.vnet.net
• Subject: [mg63350] Re: [mg63335] Questions regarding MatrixExp, and its usage
• From: Pratik Desai <pdesai1 at umbc.edu>
• Date: Sat, 24 Dec 2005 16:02:58 -0500 (EST)
• References: <BAY105-F26D2B5ABD484AE9CD7EF289A320@phx.gbl>
• Sender: owner-wri-mathgroup at wolfram.com

```Michael Chang wrote:

> Hi Pratik,
>
> Many thanks for your response and help!
>
> My only concern is about the usage of MatrixPower -- all of the
> Mathematica online documentation and examples using this function seem
> to indicate that it is only valid for an *integer* power p.
>
> Since MatrixExp[aMatrix,p] exists (and is unique) for all square
> "aMatrix" values and any *complex* value of "p", I guess that I began
> wondering under what conditions this might be equal to
>
>   MatrixPower[MatrixExp[aMatrix],p]
>
> ?  Perhaps mathematically this only holds for *integer* values of p?
> I don't know ...
>
> Anyways, many thanks again, and Happy Holidays!
>
> Regards,
>
> Michael
>
>> From: Pratik Desai <pdesai1 at umbc.edu>
To: mathgroup at smc.vnet.net
>> To: "michael_chang86 at hotmail.com" <michael_chang86 at hotmail.com>
>> Subject: [mg63350] Re: [mg63335] Questions regarding MatrixExp, and its usage
>> Date: Sat, 24 Dec 2005 09:30:52 -0500
>>
>> michael_chang86 at hotmail.com wrote:
>>
>>> Hi,
>>>
>>> For any arbitrary (possibly complex-valued) square matrix A,
>>> Mathematica enables the computation of the matrix exponential of A via
>>>
>>> In[1]:  A={{ some square matrix}};
>>> In[2]:  expA=MatrixExp[A];
>>>
>>> I was therefore wondering if
>>>
>>> MatrixExp[A p]==(MatrixExp[A]^p)
>>>
>>> where 'p' is an arbitrary complex number, and the '^' operator is my
>>> attempt to denote the matrix power, and *not* an element-by-element
>>> power for each individual matrix entry.  Or does such an expression
>>> only hold for real-valued square A matrices?  Or am I completely lost
>>> here ...?
>>>
>>> As usual, any and all help would be greatly appreciated!
>>>
>>> Regards,
>>>
>>> Michael
>>>
>>>
>>>
>> matx[A_?MatrixQ, p_]=MatrixPower[MatrixExp[A], p]
>>
>>
>> Hope this helps
>>
>> Pratik
>>
>>
>>
>>
>
>
You will never know unless you try :-)
In[10]:=
p=Random[]+Pi*I
MatrixPower[MatrixExp[IdentityMatrix[3]],p]//Chop//InputForm

Out[10]=
0.982433\[InvisibleSpace]+3.14159 \[ImaginaryI]

Out[11]//InputForm=
{{-2.670947256395083, 0, 0}, {0, -2.670947256395083, 0}, {0, 0,
-2.670947256395083}}

In[33]:=
MatrixExp[IdentityMatrix[3],p]//Chop//InputForm

Out[33]//InputForm=
{{-2.6709472563950825, 0, 0}, {0, -2.6709472563950825, 0}, {0, 0,
-2.6709472563950825}}

I think in my experience with mathematica if there are some limitation
with a particular function, the documentation always seems to highlight
it somewhere, and I did not see any explicit disclaimers regarding the
limitation for MatrixPower only working with integers. To be perfectly
honest, I don't know why In[33] works perhaps someone else on the forum
can help

Pratik

Happy Holidays to you!

PS: I hope you don't mind my posting your reply on the forum

--
Pratik Desai

```

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