Three interpretations of matrix products
linearalgebraforprogrammers.com
Three interpretations of matrix products
1–10 of 28 posts
Re: Three interpretations of matrix products
#2I really dig the integral transform one.
Re: Three interpretations of matrix products
#3I like the fourth interpretation of matrix multiplication from geometric algebra, which brilliantly encapsulates geometric transformations. This approach shifts our focus from just numbers to the geometry of space, revealing how matrices can elegantly describe rotations, reflections, and scaling. It’s a vivid and intuitive perspective that brings matrix operations to life, especially in fields where visualizing these transformations is key, like in computer graphics or physics. It’s like watching math and geometry dance together!
Re: Three interpretations of matrix products
#4I like the fourth interpretation of matrix multiplication from geometric algebra, which brilliantly encapsulates geometric transformations. This approach shifts our focus from just numbers to the geometry of space, revealing how matrices can elegantly describe rotations, reflections, and scaling. It’s a vivid and intuitive perspective that brings matrix operations to life, especially in fields where visualizing these…
The rest of your comment doesn't really seem specific to geometric algebra. You're just interpreting matrix multiplication as composition of linear maps, which is indeed a very useful and sometimes intuitive perspective.
Re: Three interpretations of matrix products
#5Gil Strang lectures are great if you want to a bit more depth.
Re: Three interpretations of matrix products
#6The first two interpretations are about matrix-vector products, and the third is about matrix-matrix product. This is a bit surprising, as I was expecting different interpretations of the same thing.
Anyway these are basic stuff, but it's nice to see them formulated!
Re: Three interpretations of matrix products
#7I like the fourth interpretation of matrix multiplication from geometric algebra, which brilliantly encapsulates geometric transformations. This approach shifts our focus from just numbers to the geometry of space, revealing how matrices can elegantly describe rotations, reflections, and scaling. It’s a vivid and intuitive perspective that brings matrix operations to life, especially in fields where visualizing these…
3blue1brown's linear algebra 'course' is excellent on this https://www.3blue1brown.com/topics/linear-algebra
Re: Three interpretations of matrix products
#8This site looks like a similar concept to https://immersivemath.com/ which is one of my favorite sites ever.
Re: Three interpretations of matrix products
#9The first two interpretations are about matrix-vector products, and the third is about matrix-matrix product. This is a bit surprising, as I was expecting different interpretations of the same thing. Anyway these are basic stuff, but it's nice to see them formulated!
Originally I had planned to add a line or two explaining how a matrix-matrix product is just a _list_ of matrix-vector products but then dropped the idea to keep it solely focused on the interpretations. I will probably add it to make it all about matrix-matrix products.
Re: Three interpretations of matrix products
#10It is also interesting how this relates to hardware implementations. Nvidia and Intel AMX appear to be using dot product engines under the hood and do a matrix multiplication in a single instruction. Apple AMX and ARM SME use outer product engines and require multiple instructions to do a single matrix multiplication.