Handling 4-block oriented matrix product and inversion in Maxima - matrix

Question

I am concerned in finding symbolic solutions and expansion to matrix products and inversions. Actually, it is something I would like to define by myself. I will explain myself.
I want to create a "mathematical" object that i will call B4MAT which represents a square matrix whose elements are 4 square half-sized matrices. So I want to define the product between two B4MAT giving me back another B4MAT whose components are calculated by applying product rules, but among matrices, not scalars.
Furthermore, and this is a very important point, consider Blockwise Inversion of a matrix. I want to define inversion of a B4MAT as an operation returning me another B4MAT whose elements are calculated using the blockwise inversion algorithm in the link.
How to achieve this in Maxima?
Thankyou

Answer 1

For the first half of your question, you just need to change matrix_element_mult to non-commutative multiplication and then use a matrix whose elements are the blocks you want. For example:
Maxima branch_5_27_base_248_ge261c5e http://maxima.sourceforge.net
using Lisp SBCL 1.0.57.0.debian
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1) A: matrix([1,2],[3,4])$ B: matrix([2,1],[3,4])$
(%i3) matrix([A,B], [B,A]);
*** output flushed ***
(%i4) C: matrix([A,B], [B,A]);
[ [ 1 2 ] [ 2 1 ] ]
[ [ ] [ ] ]
[ [ 3 4 ] [ 3 4 ] ]
(%o4) [ ]
[ [ 2 1 ] [ 1 2 ] ]
[ [ ] [ ] ]
[ [ 3 4 ] [ 3 4 ] ]
(%i5) C . C;
[ [ 5 5 ] [ 4 4 ] ]
[ [ ] [ ] ]
[ [ 18 32 ] [ 18 32 ] ]
(%o5) [ ]
[ [ 4 4 ] [ 5 5 ] ]
[ [ ] [ ] ]
[ [ 18 32 ] [ 18 32 ] ]
(%i6) matrix_element_mult: ".";
(%o6) .
(%i7) C . C;
[ [ 14 16 ] [ 13 17 ] ]
[ [ ] [ ] ]
[ [ 33 41 ] [ 33 41 ] ]
(%o7) [ ]
[ [ 13 17 ] [ 14 16 ] ]
[ [ ] [ ] ]
[ [ 33 41 ] [ 33 41 ] ]
I think you have to code up the inversion formula yourself though (don't forget you can get at the blocks with expressions like "C[1][2]" (for the top right corner) etc.