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Re: Scalars Instead of Lists with One Element

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  • Subject: [mg83565] Re: Scalars Instead of Lists with One Element
  • From: Szabolcs Horvát <szhorvat at>
  • Date: Fri, 23 Nov 2007 05:29:10 -0500 (EST)
  • References: <fhu75c$72j$> <> <fi3jfv$b34$>

Gregory Lypny wrote:
> Thanks everyone for your insights,
> I've found the problem.  Say you want to sum a list whose elements are  
> 13 and 9.  Mathematica will return a list with one element, {22},  
> rather than 22 if the original list is specified as a 2x1 column vector.
> x = {{13}, {9}}; y = Total@x    >>> returns {22}
> This also happens if you write x as I have above, or you use  
> Mathematica's Insert menu to create a more visually appealing column  
> vector,

Tip:  Use CTRL+Enter and CTRL+, instead of the insert menu.

> but it does not happen if you define x as a 2x1 array using  
> the Array command.

Yes, it does:

In[1]:= Array[# &, {2, 1}]
Out[1]= {{1}, {2}}

In[2]:= Total[%]
Out[2]= {3}

> I think it will happen with any matrix calculation whose result should  
> otherwise be a scalar.  If we now let y be the row vector {1, 1} then
> y.x    >>> returns {22}

This is because Mathematica works with tensors of arbitrary dimensions, 
not just matrices or vectors.  So in Mathematica it does not make sense 
to speak about column or row vectors---all vectors (1D arrays) are 
treated in the same way.  {{1},{2}} is a 2 by 1 matrix and {{1, 2}} is a 
1 by 2 matrix, not vectors.  (Of course all this is just a question of 
naming conventions.  I am just trying to explain why Mathematica works 
this way.)

> The upshot of this is that any table that is created from calculations  
> that make use of column vectors or matrix math will likely have a  
> depth greater than 3, and you won't be able to cut and paste directly  
> into a word processor or spreadsheet.

When you multiply a vector with a matrix, you get a vector.  If you 
multiply a vector with a vector, you get a scalar:

In[4]:= {x, y}.{{a, b}, {c, d}}
Out[4]= {a x + c y, b x + d y}

In[5]:= {{a, b}, {c, d}}.{x, y}
Out[5]= {a x + b y, c x + d y}

In[6]:= {x, y}.{a, b}
Out[6]= a x + b y

If the distinction between column and row vectors is important, just 
work with matrices, and you'll always get matrices as the result (no 

In[7]:= {{x, y}}.{{a}, {b}}
Out[7]= {{a x + b y}}

In[8]:= {{a}, {b}}.{{x, y}}
Out[8]= {{a x, a y}, {b x, b y}}

If it is not important to differentiate between row and column vectors, 
then use 1D arrays to represent vectors.  The Outer product can be 
calculated like this:

In[9]:= Outer[Times, {a, b}, {x, y}]
Out[9]= {{a x, a y}, {b x, b y}}

> I'm going to have a look at some of the work-arounds that have been  
> suggested in this thread and my related thread "Copy and Pasting  
> Tables into Spreadsheet".
> Gregory


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