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Re: Multiply 2 matrices where one contains differential

  • To: mathgroup at smc.vnet.net
  • Subject: [mg104468] Re: [mg104417] Multiply 2 matrices where one contains differential
  • From: DrMajorBob <btreat1 at austin.rr.com>
  • Date: Sun, 1 Nov 2009 04:01:05 -0500 (EST)
  • References: <20091029225146.K51SM.569239.imail@eastrmwml30>
  • Reply-to: drmajorbob at yahoo.com

You could also simply multiply A*B, if A and B are the same dimensions.

Anyway... I gave a solution. If somebody has a better one, we'll see.

Bobby

On Sat, 31 Oct 2009 16:52:04 -0500, Nasser M. Abbasi <nma at 12000.org> wrote:

> DrMajorBob;
>
> Thanks for your detailed reply, I need to look at your code more  
> closely. But just to answer your initial point
>
>> That doesn't look like a Dot product at all, since it operates a 3x2  
>> matrix on a 2x3 and gets a 3x3.
>
> Yes. I am doing matrix products ofcourse, so 3x2 matrix when multiplied  
> by 2x3 matrix better result in 3x3 or else something is really wrong :)
>
> From help on Dot
> "The standard (built-in) usage still exists a.b.c or Dot[a, b, c] gives  
> products of vectors, matrices and tensors"
>
> I use A.B to multiply matrices by each others? may be I am missing your  
> point here.  How else would you multiply matrices with each others in  
> Mathematica other than using the Dot?
>
> Best,
> --Nasser
>
>
> ----- Original Message ----- Subject: Re: [mg104417] Multiply 2 matrices  
> where one contains differential operators with one that contains  
> functions of x and y
>
>
>> That doesn't look like a Dot product at all, since it operates a 3x2  
>> matrix on a 2x3 and gets a 3x3.
>>
>> So... to find out what's really happening:
>>
>> a = Array[f, {3, 2}]
>> b = Array[g, {2, 3}]
>> {rowsA, colsA} = Dimensions[a];
>> {rowsB, colsB} = Dimensions[b];
>> r = Table[0, {rowsA}, {colsB}];(*where the result of A.B goes*)For[
>>  i = 1, i <= rowsA, i++,
>>  For[j = 1, j <= colsB, j++,
>>   For[ii = 1, ii <= rowsB, ii++,
>>    r[[i, j]] = r[[i, j]] + a[[i, ii]] /@ {b[[ii, j]]}]]]
>> r
>> Dimensions@r
>>
>> {{f[1, 1], f[1, 2]}, {f[2, 1], f[2, 2]}, {f[3, 1], f[3, 2]}}
>>
>> {{g[1, 1], g[1, 2], g[1, 3]}, {g[2, 1], g[2, 2], g[2, 3]}}
>>
>> {{{f[1, 1][g[1, 1]] + f[1, 2][g[2, 1]]}, {f[1, 1][g[1, 2]] +
>>     f[1, 2][g[2, 2]]}, {f[1, 1][g[1, 3]] +
>>     f[1, 2][g[2, 3]]}}, {{f[2, 1][g[1, 1]] +
>>     f[2, 2][g[2, 1]]}, {f[2, 1][g[1, 2]] +
>>     f[2, 2][g[2, 2]]}, {f[2, 1][g[1, 3]] +
>>     f[2, 2][g[2, 3]]}}, {{f[3, 1][g[1, 1]] +
>>     f[3, 2][g[2, 1]]}, {f[3, 1][g[1, 2]] +
>>     f[3, 2][g[2, 2]]}, {f[3, 1][g[1, 3]] + f[3, 2][g[2, 3]]}}}
>>
>> {3, 3, 1}
>>
>> That looks like an outer product of the rows of a with the columns of b.
>>
>> Here's a somewhat similar structure (with one less low-level dimension):
>>
>> Outer[h, a, Transpose@b, 1, 1]
>> Dimensions@%
>>
>> {{h[{f[1, 1], f[1, 2]}, {g[1, 1], g[2, 1]}],
>>   h[{f[1, 1], f[1, 2]}, {g[1, 2], g[2, 2]}],
>>   h[{f[1, 1], f[1, 2]}, {g[1, 3], g[2, 3]}]}, {h[{f[2, 1],
>>     f[2, 2]}, {g[1, 1], g[2, 1]}],
>>   h[{f[2, 1], f[2, 2]}, {g[1, 2], g[2, 2]}],
>>   h[{f[2, 1], f[2, 2]}, {g[1, 3], g[2, 3]}]}, {h[{f[3, 1],
>>     f[3, 2]}, {g[1, 1], g[2, 1]}],
>>   h[{f[3, 1], f[3, 2]}, {g[1, 2], g[2, 2]}],
>>   h[{f[3, 1], f[3, 2]}, {g[1, 3], g[2, 3]}]}}
>>
>> {3, 3}
>>
>> It remains to figure out what "h" should be. A simple solution is
>>
>> h[{a_, b_}, {c_, d_}] := a[c] + b[d]
>> Outer[h, a, Transpose@b, 1, 1]
>> Dimensions@%
>>
>> {{f[1, 1][g[1, 1]] + f[1, 2][g[2, 1]],
>>   f[1, 1][g[1, 2]] + f[1, 2][g[2, 2]],
>>   f[1, 1][g[1, 3]] + f[1, 2][g[2, 3]]}, {f[2, 1][g[1, 1]] +
>>    f[2, 2][g[2, 1]], f[2, 1][g[1, 2]] + f[2, 2][g[2, 2]],
>>   f[2, 1][g[1, 3]] + f[2, 2][g[2, 3]]}, {f[3, 1][g[1, 1]] +
>>    f[3, 2][g[2, 1]], f[3, 1][g[1, 2]] + f[3, 2][g[2, 2]],
>>   f[3, 1][g[1, 3]] + f[3, 2][g[2, 3]]}}
>>
>> {3, 3}
>>
>> To get back the low-level (probably unnecessary) dimension:
>>
>> h[{a_, b_}, {c_, d_}] := {a[c] + b[d]}
>> Outer[h, a, Transpose@b, 1, 1]
>> Dimensions@%
>>
>> {{{f[1, 1][g[1, 1]] + f[1, 2][g[2, 1]]}, {f[1, 1][g[1, 2]] +
>>     f[1, 2][g[2, 2]]}, {f[1, 1][g[1, 3]] +
>>     f[1, 2][g[2, 3]]}}, {{f[2, 1][g[1, 1]] +
>>     f[2, 2][g[2, 1]]}, {f[2, 1][g[1, 2]] +
>>     f[2, 2][g[2, 2]]}, {f[2, 1][g[1, 3]] +
>>     f[2, 2][g[2, 3]]}}, {{f[3, 1][g[1, 1]] +
>>     f[3, 2][g[2, 1]]}, {f[3, 1][g[1, 2]] +
>>     f[3, 2][g[2, 2]]}, {f[3, 1][g[1, 3]] + f[3, 2][g[2, 3]]}}}
>>
>> {3, 3, 1}
>>
>> For your actual problem, that's
>>
>> a = {{D[#1, x] &, 0 &}, {0 &, D[#1, y] &}, {D[#1, y] &, D[#1, x] &}};
>> b = {{x*y, x^3*y, 3*x + y^2}, {2*x, x^4*y, y^2}};
>> h[{a_, b_}, {c_, d_}] := {a[c] + b[d]}
>> Outer[h, a, Transpose@b, 1, 1]
>>
>> {{{y}, {3 x^2 y}, {3}}, {{0}, {x^4}, {2 y}}, {{2 + x}, {x^3 +
>>     4 x^3 y}, {2 y}}}
>>
>> which agrees with your nested For method.
>>
>> The function h could be done differently, since it's analogous to
>>
>> Inner[Compose, {aa, bb}, {c, d}, Plus]
>>
>> aa[c] + bb[d]
>>
>> For instance:
>>
>> a = {{D[#1, x] &, 0 &}, {0 &, D[#1, y] &}, {D[#1, y] &, D[#1, x] &}};
>> b = {{x*y, x^3*y, 3*x + y^2}, {2*x, x^4*y, y^2}};
>> h = Inner[Compose, ##, Plus] &;
>> Outer[h, a, Transpose@b, 1, 1]
>>
>> {{y, 3 x^2 y, 3}, {0, x^4, 2 y}, {2 + x, x^3 + 4 x^3 y, 2 y}}
>>
>> That omits the low-level List, but it can be added in a hundred  
>> different ways. For instance,
>>
>> a = {{D[#1, x] &, 0 &}, {0 &, D[#1, y] &}, {D[#1, y] &, D[#1, x] &}};
>> b = {{x*y, x^3*y, 3*x + y^2}, {2*x, x^4*y, y^2}};
>> h = Inner[Compose, ##, Plus] &;
>> Map[List, Outer[h, a, Transpose@b, 1, 1], {2}]
>>
>> {{{y}, {3 x^2 y}, {3}}, {{0}, {x^4}, {2 y}}, {{2 + x}, {x^3 +
>>     4 x^3 y}, {2 y}}}
>>
>> Bobby
>>
>> On Sat, 31 Oct 2009 01:50:39 -0500, Nasser M. Abbasi <nma at 12000.org>  
>> wrote:
>>
>>> Hello,
>>> Version 7
>>>
>>> Lets say A is a 3 by 2 matrix, which contains differential operators  
>>> in some
>>> entries and 0 in all other entries, as in
>>>
>>> A= {  {  d/dx ,  0  } ,  {0  ,  d/dy } ,  {  d/dy ,  d/dx }  }
>>>
>>> And I want to multiply the above with say a 2 by 3 matrix whose  
>>> entries are
>>> functions of x and y as in
>>>
>>> B = {{x*y,  x^3*y,  3*x + y^2}, {2*x,  x^4*y,  y^2}}
>>>
>>> I'd like to somehow be able to do A.B, but ofcourse here I can't, as I  
>>> need
>>> to "apply" the operator on each function as the matrix multiplication  
>>> is
>>> being carried out.
>>>
>>> I tried to somehow integrate applying the operators in A into the  
>>> matrix
>>> multiplication of A by B, but could not find a short "functional" way.
>>>
>>> So I ended up solving this by doing the matrix multiplication by hand  
>>> using
>>> for loops (oh no) so that I can 'get inside' the loop and be able to  
>>> apply
>>> the operator to each entry. This is my solution:
>>>
>>>
>>> A = {{D[#1, x] & , 0 & }, {0 & , D[#1, y] & },  {D[#1, y] & , D[#1, x]  
>>>   }}
>>> B = {{x*y, x^3*y, 3*x + y^2}, {2*x, x^4*y, y^2}}
>>>
>>> {rowsA, colsA} = Dimensions[A];
>>> {rowsB, colsB} = Dimensions[B];
>>>
>>> r = Table[0, {rowsA}, {colsB}]; (*where the result of A.B goes *)
>>>
>>> For[i = 1, i <= rowsA, i++,
>>>    For[j = 1, j <= colsB, j++,
>>>        For[ii = 1, ii <= rowsB, ii++,
>>>               r[[i,j]] = r[[i,j]] + A[[i,ii]] /@ {B[[ii,j]]}
>>>              ]
>>>        ]
>>>  ]
>>>
>>> MatrixForm[r]
>>>
>>> The above work, but I am sure a Mathematica expert here can come up  
>>> with a
>>> true functional solution or by using some other Mathematica function  
>>> which I
>>> overlooked to do the above in a more elegent way.
>>>
>>> --Nasser
>>>
>>>
>>
>>
>> -- DrMajorBob at yahoo.com
>


-- 
DrMajorBob at yahoo.com


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