Re: A question about Axiom capabilities

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I just wanted to thank you all for taking this task seriously. I've been
looking for some CAS that provided significant symbolic matrix
manipulations, including matrix calculus. I found other tools that work
with tensors and provide tensor calculus (Maxima CAS has 3 tensor
packages), but second order tensors and matrices are not quite the same
thing. Furthermore, the packages were so general that they lacked specific
rules (derivatives of determinants, inverses, etc.) which worked fine on
matrices.

So once again, just saying thanks.



On Thu, Feb 28, 2013 at 7:46 PM, u1204 <[email protected]> wrote:

> Raoul,
>
> If you look at the matrixcookbook that Mike mentioned, the first 10
> equations are:
>
>    (A*B)^-1      = B^-1 * A^-1
>    (A*B*C...)^-1 = ...C^-1 * B^-1 * A^-1
>    (A^T)^-1      = (A^-1)^T
>    (A+B)^T       = A^T + B^T
>    (A*B)^T       = B^T * A^T
>    (A*B*C...)^T  = C^T * B^T *A^T
>    (A^H)^-1      = (A^-1)^H
>    (A+B)^H       = A^H + B^H
>    (A*B)^H       = B^H * A^H
>    (A*B*C...)^H  = ...C^H * B^H * A^H
>
> It seems like we could create a SymbolicMatrix algebra that could
> perform these manipulations with uninterpreted matrix symbols
> A, B, C with a special recognized symbol 'T'.
>
> These matrices could have actual values which, for certain operations
> are ignored, so that
>
>        (A*B)^-1      = B^-1 * A^-1
>
> but for other operations would be evaluated as in:
>
>        eval(B^-1 * A^-1)
>
> giving the actual matrix result shown element by element.
>
> An additional enhancement would be to make a SymbolicMatrixCategory
> so that there could be specific domains such as
>
>    GeneralSymbolicMatrix,
>    SymmetricSymbolicMatrix,
>    UpperTriangularSymbolicMatrix,
>    DiagonalSymbolicMatrix,
>
> etc which could exploit certain matrix-level properties at the
> symbolic level.
>
> This isn't exactly what Mike was originally asking but I think
> that Axiom ought to be able to symbolically compute the equations
> in the handbook.
>
> Tim Daly
>
>
>

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<div>I just wanted to thank you all for taking this task seriously. I&#39;v=
e been looking for some CAS that provided significant symbolic matrix manip=
ulations, including matrix calculus. I found other tools that work with ten=
sors and provide tensor calculus (Maxima CAS has 3 tensor packages), but se=
cond order tensors and matrices are not quite the same thing. Furthermore, =
the packages were so general that they lacked specific rules (derivatives o=
f determinants, inverses, etc.) which worked fine on matrices.</div>
<div><br></div><div>So once again, just saying thanks.</div><div><div><br><=
/div><div><br></div><div><br><div class=3D"gmail_quote">On Thu, Feb 28, 201=
3 at 7:46 PM, u1204 <span dir=3D"ltr">&lt;<a href=3D"mailto:daly@axiom-deve=
loper.org" target=3D"_blank">[email protected]</a>&gt;</span> wrote:=
<br>
<blockquote class=3D"gmail_quote" style=3D"margin:0 0 0 .8ex;border-left:1p=
x #ccc solid;padding-left:1ex">Raoul,<br>
<br>
If you look at the matrixcookbook that Mike mentioned, the first 10<br>
equations are:<br>
<br>
=A0 =A0(A*B)^-1 =A0 =A0 =A0=3D B^-1 * A^-1<br>
=A0 =A0(A*B*C...)^-1 =3D ...C^-1 * B^-1 * A^-1<br>
=A0 =A0(A^T)^-1 =A0 =A0 =A0=3D (A^-1)^T<br>
=A0 =A0(A+B)^T =A0 =A0 =A0 =3D A^T + B^T<br>
=A0 =A0(A*B)^T =A0 =A0 =A0 =3D B^T * A^T<br>
=A0 =A0(A*B*C...)^T =A0=3D C^T * B^T *A^T<br>
=A0 =A0(A^H)^-1 =A0 =A0 =A0=3D (A^-1)^H<br>
=A0 =A0(A+B)^H =A0 =A0 =A0 =3D A^H + B^H<br>
=A0 =A0(A*B)^H =A0 =A0 =A0 =3D B^H * A^H<br>
=A0 =A0(A*B*C...)^H =A0=3D ...C^H * B^H * A^H<br>
<br>
It seems like we could create a SymbolicMatrix algebra that could<br>
perform these manipulations with uninterpreted matrix symbols<br>
A, B, C with a special recognized symbol &#39;T&#39;.<br>
<br>
These matrices could have actual values which, for certain operations<br>
are ignored, so that<br>
<br>
=A0 =A0 =A0 =A0(A*B)^-1 =A0 =A0 =A0=3D B^-1 * A^-1<br>
<br>
but for other operations would be evaluated as in:<br>
<br>
=A0 =A0 =A0 =A0eval(B^-1 * A^-1)<br>
<br>
giving the actual matrix result shown element by element.<br>
<br>
An additional enhancement would be to make a SymbolicMatrixCategory<br>
so that there could be specific domains such as<br>
<br>
=A0 =A0GeneralSymbolicMatrix,<br>
=A0 =A0SymmetricSymbolicMatrix,<br>
=A0 =A0UpperTriangularSymbolicMatrix,<br>
=A0 =A0DiagonalSymbolicMatrix,<br>
<br>
etc which could exploit certain matrix-level properties at the<br>
symbolic level.<br>
<br>
This isn&#39;t exactly what Mike was originally asking but I think<br>
that Axiom ought to be able to symbolically compute the equations<br>
in the handbook.<br>
<br>
Tim Daly<br>
<br>
<br>
</blockquote></div><br></div></div>

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