A Programmer’s Intuition for Matrix Multiplication
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A Programmer’s Intuition for Matrix Multiplication
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Re: A Programmer’s Intuition for Matrix Multiplication
#2I'd prefer stating that matrices actually don't mean anything in isolation. They are simply representations of linear maps. What happens to the elements of a matrix before the elements are formed is irrelevant to the matrix itself.
..And to be clear, I am not a purist.
Re: A Programmer’s Intuition for Matrix Multiplication
#3I'd prefer stating that matrices actually don't mean anything in isolation. They are simply representations of linear maps. What happens to the elements of a matrix before the elements are formed is irrelevant to the matrix itself. ..And to be clear, I am not a purist.
Purist test: aren't vectors just Nx1 matrices?
Joking aside, matrices are linear maps in the context of multiplication. They can absolutely have meanings assigned outside the usual "when multiplied by a vector" type of calculations. For example the "adjacency matrix" representation of a graph is very often never multiplied by a vector even in academic algorithm descriptions. YMMV if you call it a (lookup) table in that context but the term "matrix" seems fairly well established there.
Re: A Programmer’s Intuition for Matrix Multiplication
#4I'd prefer stating that matrices actually don't mean anything in isolation. They are simply representations of linear maps. What happens to the elements of a matrix before the elements are formed is irrelevant to the matrix itself. ..And to be clear, I am not a purist.
Purist test: aren't vectors just Nx1 matrices? Joking aside, matrices are linear maps in the context of multiplication. They can absolutely have meanings assigned outside the usual "when multiplied by a vector" type of calculations. For example the "adjacency matrix" representation of a graph is very often never multiplied by a vector even in academic algorithm descriptions. YMMV if you call it a (lookup) table in th…
You can do many linear algebra operations on the adjacency matrix. It's a proper matrix, in the full sense, not just a table of numbers.
Re: A Programmer’s Intuition for Matrix Multiplication
#5I'd prefer stating that matrices actually don't mean anything in isolation. They are simply representations of linear maps. What happens to the elements of a matrix before the elements are formed is irrelevant to the matrix itself. ..And to be clear, I am not a purist.
Purist test: aren't vectors just Nx1 matrices? Joking aside, matrices are linear maps in the context of multiplication. They can absolutely have meanings assigned outside the usual "when multiplied by a vector" type of calculations. For example the "adjacency matrix" representation of a graph is very often never multiplied by a vector even in academic algorithm descriptions. YMMV if you call it a (lookup) table in th…
And covectors 1xN matrices?
Re: A Programmer’s Intuition for Matrix Multiplication
#6I'd prefer stating that matrices actually don't mean anything in isolation. They are simply representations of linear maps. What happens to the elements of a matrix before the elements are formed is irrelevant to the matrix itself. ..And to be clear, I am not a purist.
Purist test: aren't vectors just Nx1 matrices? Joking aside, matrices are linear maps in the context of multiplication. They can absolutely have meanings assigned outside the usual "when multiplied by a vector" type of calculations. For example the "adjacency matrix" representation of a graph is very often never multiplied by a vector even in academic algorithm descriptions. YMMV if you call it a (lookup) table in th…
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Re: A Programmer’s Intuition for Matrix Multiplication
#7I'd prefer stating that matrices actually don't mean anything in isolation. They are simply representations of linear maps. What happens to the elements of a matrix before the elements are formed is irrelevant to the matrix itself. ..And to be clear, I am not a purist.
Purist test: aren't vectors just Nx1 matrices? Joking aside, matrices are linear maps in the context of multiplication. They can absolutely have meanings assigned outside the usual "when multiplied by a vector" type of calculations. For example the "adjacency matrix" representation of a graph is very often never multiplied by a vector even in academic algorithm descriptions. YMMV if you call it a (lookup) table in th…
> They can absolutely have meanings assigned outside the usual "when multiplied by a vector" type of calculations. For example the "adjacency matrix" representation of a graph is very often never multiplied by a vector even in academic algorithm descriptions.
Not disputing your basic point, but I'm not sure "very often never" is entirely accurate. In the GraphBLAS API vector/matrix and matrix/vector multiplication is key for almost all algorithms that depend on breadth first search, this is because matrix/vector multiplication is a single step in a BFS which is the core operation of the GraphBLAS. The vector is typically used to hold the current frontier (and mask). See the "Shortest Path Lengths" example here:
https://github.com/michelp/pygraphblas/blob/master/pygraphbl...
(disclosure, I am the pygraphblas author).
> Purist test: aren't vectors just Nx1 matrices?
I like it. :) As an interesting point, the GraphBLAS defines both Matrix and Vector types, but internally in the SuiteSparse implementation, they are all just unified into one type and yes, vectors are Nx1 or 1xN.
Re: A Programmer’s Intuition for Matrix Multiplication
#8I'd prefer stating that matrices actually don't mean anything in isolation. They are simply representations of linear maps. What happens to the elements of a matrix before the elements are formed is irrelevant to the matrix itself. ..And to be clear, I am not a purist.
Purist test: aren't vectors just Nx1 matrices? Joking aside, matrices are linear maps in the context of multiplication. They can absolutely have meanings assigned outside the usual "when multiplied by a vector" type of calculations. For example the "adjacency matrix" representation of a graph is very often never multiplied by a vector even in academic algorithm descriptions. YMMV if you call it a (lookup) table in th…
> the "adjacency matrix" representation of a graph is very often never multiplied by a vector even in academic algorithm descriptions.
Not true. There is a whole field (graph signal processing) that studies graphs in terms of their description in matrix form (adjacency, connectivity and more). For example, you can find connected components by looking at the eigenvectors of a certain matrix.
Re: A Programmer’s Intuition for Matrix Multiplication
#9I'd prefer stating that matrices actually don't mean anything in isolation. They are simply representations of linear maps. What happens to the elements of a matrix before the elements are formed is irrelevant to the matrix itself. ..And to be clear, I am not a purist.
Purist test: aren't vectors just Nx1 matrices? Joking aside, matrices are linear maps in the context of multiplication. They can absolutely have meanings assigned outside the usual "when multiplied by a vector" type of calculations. For example the "adjacency matrix" representation of a graph is very often never multiplied by a vector even in academic algorithm descriptions. YMMV if you call it a (lookup) table in th…
Yes, they pretty much are unless you're in the business of linear codes—then they're actually 1xN matrices. A pretty annoying convention for someone coming in from the general algebra: takes a time to stop mentally transposing everything.
Re: A Programmer’s Intuition for Matrix Multiplication
#10I'd prefer stating that matrices actually don't mean anything in isolation. They are simply representations of linear maps. What happens to the elements of a matrix before the elements are formed is irrelevant to the matrix itself. ..And to be clear, I am not a purist.
Purist test: aren't vectors just Nx1 matrices? Joking aside, matrices are linear maps in the context of multiplication. They can absolutely have meanings assigned outside the usual "when multiplied by a vector" type of calculations. For example the "adjacency matrix" representation of a graph is very often never multiplied by a vector even in academic algorithm descriptions. YMMV if you call it a (lookup) table in th…
> For example the "adjacency matrix" representation of a graph is very often never multiplied by a vector even in academic algorithm descriptions.
Unless you're doing spectral graph theory, which looks at properties of the eigenvalues and eigenvectors of the graph's adjacency matrix. Even then, you don't usually multiply the adjacency matrix by a vector explicitly.