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MathGroup Archive 2006

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Re: mathematica newbie question

  • To: mathgroup at smc.vnet.net
  • Subject: [mg66912] Re: mathematica newbie question
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Sat, 3 Jun 2006 03:26:19 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <e5osv8$i51$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Jeremy Watts wrote:
> Hi,
> 
> Just trying to get to grips with Mathematica and matrices.  Why is it when I
> enter :-
> {0, 0, 0}, {1, -1, -1}, {-1, 1, 1}}^2
> 
> then Mathematica returns :-
> 
> {{0, 0, 0}, {1, 1, 1}, {1, 1, 1}}
> 
> and not {{0,0,0},{0,0,0},{0,0,0}} (the actual square of the matrix) as I'd
> have expected?
> 
> Is it treating what I entered as separate vectors or something, and not  an
> actual matrix?


Hi Jeremy,

Mathematica knows nothing about "matrices" per se. Well, this statement 
is slightly extreme; but the fact is that, to Mathematica, a list of 
lists is first a list of lists that can contains lists of different 
lengths, each of them holding elements of different types. For instance, 
the following list is a valid construct in Mathematica

{{a,b,c},{"Some string",{1,2,{3,4},index}},{Sin,Cos}}

Of course, this expression cannot be interpreted as a matrix.

On the other hand, your original expression -- called "mat" in the 
remaining of this post, see In[1] -- can be interpreted as a matrix 
since, say, taking the dot product of a 3 by 3 structure of integers 
makes sense (see In[4]). However, mat is still a regular expression and 
Mathematica can works on it element by element (see In[2] and In[3]).

Therefore, to get raise a matrix to a power, either use repeatedly the 
dot product or use special built-in functions such as MatrixExp or 
MatrixPower (see In[4] and In[5]).

Finally, the article "Functions of Matrices" [1] may be of interest.

In[1]:=
mat = {{0, 0, 0}, {1, -1, -1}, {-1, 1, 1}};

In[2]:=
mat^2

Out[2]=
{{0, 0, 0}, {1, 1, 1}, {1, 1, 1}}

In[3]:=
mat*mat

Out[3]=
{{0, 0, 0}, {1, 1, 1}, {1, 1, 1}}

In[4]:=
mat . mat

Out[4]=
{{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}

In[5]:=
MatrixPower[mat, 2]

Out[5]=
{{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}

HTH,
Jean-Marc

[1] 
http://documents.wolfram.com/mathematica/Built-inFunctions/AdvancedDocumentation/LinearAlgebra/LinearAlgebraInMathematica/MatrixComputations/AdvancedDocumentationLinearAlgebra3.5.html


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