Why not arrange the matrix elements in triangular lattice, or other other topologies?
In my opinion, the strength and beauty of Matrix theory do not stem as much from the arrangement structure (rectangular array) than the ingenious rules developed by mathematicians over centuries to manipulate them, transforming them into one another while preserving one or more attributes of the array, for example, the trace and determinant preservation under similarity transformation, inertia under congruence transformation and the more generally, the Equivalence transformations which only preserve the rank of the matrix.
an array is just a way of visualizing a set of numbers, it comes to life through the rules of composition/manipulation defined on them, that generates the Array Algebra.
I suppose you could also have a rectangular block in 3D.
It would multiply a matrix (same size as a face) to give another matrix.
The useful product for matrices is defined as row by column product, because it corresponds to composition of linear maps, so the rectangular array is simply the most convenient.
I liked this joke-question! Best! Like this - why "zero" means "nothing"??? Fantastically!
I used matrices any topologies (circle, ellipse, and with chaotic outlines ) in my book on pages 437- 438
Book Fundamentals of Military-Technical Research: Theory and Appl...
Such matrix topologies need me to build of the model of digital antenna arrays.
As the main operation for such matrices, I used the penetration face product:
Article A family of face products of matrices and its properties
and generalized face product:
Article Generalized face-products of matrices in models of digital a...
In my opinion, circle matrices can be useful in Convolution Neural Networks as a mask for filtering input data matrix.
I think that in as the entries of matrices are arranged in a rectangular way, triangular arrangement can also be done by it, but the converse. For instance you can consider upper or lower triangular matrices.
- Badges
- Science topic
- Similar topics
- Mathematics
- Algebra
- Matrix