AxB=B ?

  1. AxB=B ????

    Hello, This my first post .. I hope to find someone can help me.

    My question:
    Assume that we have a 3x3 matrix B, is there a matrix A such that

    AxB=B

    ??

    Thanks
     
  2. jcsd
  3. micromass

    micromass 19,229
    Staff Emeritus
    Science Advisor
    Education Advisor

    Re: AxB=B ????

    Take A the identity matrix. That is, take

    [tex]A=\left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{array}\right)[/tex]
     
  4. Re: AxB=B ????

    I tried that .. it doesnt work for cross product !!
     
  5. Re: AxB=B ????

    Its not a multiplication .. its a cross product. For example:

    A =

    1 2 3
    4 5 6
    7 8 9

    >> B=eye(3)

    B =

    1 0 0
    0 1 0
    0 0 1

    >> cross(B,A)

    ans =

    0 8 -6
    -7 0 3
    4 -2 0

    ???
     
  6. micromass

    micromass 19,229
    Staff Emeritus
    Science Advisor
    Education Advisor

    Re: AxB=B ????

    So, how do you define the cross product for matrices??
     
  7. D H

    Staff: Mentor

    Re: AxB=B ????

    I would double check on that if I were you. The standard definition of A×B is that of matrix multiplication:

    [tex](A\times B)_{ij} = \sum_k A_{ik} B_{kj}[/tex]

    Interpreting A×B to mean Matlab's cross(A,B) when A and B are matrices is extremely non-standard.
     
  8. Re: AxB=B ????

    Cross products for matrices can be done in Matlab, and it seems to be defined as several cross products in the matrix. Are you sure this is what you want? I have never seen this before. AxB has always been ordinary matrix multiplication.
     
  9. Re: AxB=B ????

    @Yazid: I'm not sure there is even a sensible way to define a cross product for matrices in 3 dimensions. The closest you can probably get to it is just the matrix commutator. It's a antisymmetric product that takes two matrices and returns a matrix.

    Then [A,B] = B is basically an eigenvalue problem for the adjoint representation of A...
    where you've been given an eigenvector (B) and need to find an operator (ad(A)) with that eigenvector.
     
  10. Re: AxB=B ????

    Actually, matlab does cross products on arrays of 3d vectors.
    It works something like (I don't have matlab, so I can't be sure)

    So if A = (a1,a2,...,an) and B=(b1,b2,...,bn) with ai and bi 3d vectors, then

    cross(A,B) = ( cross(a1,b1), cross(a2,b2), ..., cross(an,bn) )

    In Mathematica, the example provided can be written as

    Code (Text):

    A = {{1, 2, 3},
         {4, 5, 6},
         {7, 8, 9}};
    B = IdentityMatrix[3];

    Transpose[Table[Cross[Transpose[B][[i]], Transpose[A][[i]]], {i, 3}]]
    (* Output:
    {{ 0,  8, -6},
     {-7,  0,  3},
     { 4, -2,  0}}
    *)
     
    So for n=3, the original problem becomes

    cross(A,B)=B ==> (a1xb1, a2xb2, a3xb3) = (b1, b2, b3)

    which requires that aixbi=bi, something that is never true...
     
  11. pwsnafu

    pwsnafu 931
    Science Advisor

    Re: AxB=B ????

    I'd use the Thread command myself.
     
  12. Re: AxB=B ????

    @pwsnafu

    I couldn't get Thread to work without Cross complaining about its arguments - although the error messages can be suppressed. Overall, I found that a combination of Apply at level 1 (@@@) the fastest, but not by much. Table is definitely the clearest to someone who is not used to functional programming.

    Here's some timings using my custom TimeAv function. The FullForm of Transpose, instead of its typeset form, definitely makes the whole thing less readable...

    Code (Text):

    In[1]:= A = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}};
            B = IdentityMatrix[3];

    In[3]:= SetOptions[TimeAv, Method -> {"MinNum", 10000}, "BlockSize" -> 1000];

    In[4]:= Transpose[Table[Cross[Transpose[B][[i]], Transpose[A][[i]]],
                      {i, 3}]] // TimeAv

    During evaluation of In[4]:= Total wall time is 4.817133,
    total cpu time is 4.07 and total time spent evaluating the expression is 4.07

    During evaluation of In[4]:= The expression was evaluated 10000 times,
    in blocks of 1000 runs. This yields a mean timing of 0.000407
    with a blocked standard deviation of 7.81025*10^-6.

    Out[4]= {0.000407, {{0, 8, -6}, {-7, 0, 3}, {4, -2, 0}}}

    In[5]:= Transpose[Apply[Cross, Transpose[{Transpose[B], Transpose[A]}],
                            {1}]] // TimeAv

    During evaluation of In[5]:= Total wall time is 3.847493,
    total cpu time is 3.85 and total time spent evaluating the expression is 3.85

    During evaluation of In[5]:= The expression was evaluated 10000 times,
    in blocks of 1000 runs. This yields a mean timing of 0.000385
    with a blocked standard deviation of 5.*10^-6.

    Out[5]= {0.000385, {{0, 8, -6}, {-7, 0, 3}, {4, -2, 0}}}

    In[6]:= Quiet[Transpose[Thread[Cross[Transpose[B], Transpose[A]]]],
                  Cross::nonn1] // TimeAv

    During evaluation of In[6]:= Total wall time is 4.462055,
    total cpu time is 4.46 and total time spent evaluating the expression is 4.46

    During evaluation of In[6]:= The expression was evaluated 10000 times,
    in blocks of 1000 runs. This yields a mean timing of 0.000446
    with a blocked standard deviation of 9.16515*10^-6.

    Out[6]= {0.000446, {{0, 8, -6}, {-7, 0, 3}, {4, -2, 0}}}
     
     
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