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I have a stupid question. I still don't really fully understand why matrix multiplication came to be row by column. I mean you could define things as row by tog

I have a stupid question. I still don't really fully understand why matrix multiplication came to be row by column. I mean you could define things as row by together with the transpose operator right? So why is row by column so obviously the right way to define multiplication?

ecnahc515 4y ago
It's probably related to how we read and write. Top down, left to right. Rows are the entries, columns are attributes. Your list will potentially grow forever, whereas the attributes are typically fixed. Your width is typically fixed, but you can continue writing top down and continue on a new sheet/etc, as you add more rows. It just makes sense with how we typically organize information on pen/paper.
tptacek 4y ago
I'm bad at math, and know some linear algebra only because it's useful in cryptanalysis, but the way I keep this in my head is that matrix multiplication, mechanically, is just repeated application of matrix-vector multiplication; you're just doing multiple columns, not just one. And then, matrix-column vector multiplication is linear function application, right? And you can do column-times-row too, right? It's something like the sum of multiple outer products? (These question marks are all genuine! I probably have most of this wrong!)
theteapot 4y ago
It's a dark pattern to make make Matrix math seem more complicated than it is.
geokon 4y ago
The way things are you have a nice set of properties which allow you to manipulate system of matrix equations - but it's just notation and convention. You could do column by row as well - it'd be all the same math just effectively transposed. If you defined matrix multiplication as row times row then you are either limited to square matrices or you loose some of the convenient properties of matrix multiplication ABC = A(BC) = (AB)C (AB)^T = B^T A^T and most importantly AB =/= BA
ndriscoll 4y ago
Vectors are written as columns. If I have a linear function, I can describe it to you by just telling you what it does to each basis element, so that's what a matrix is: a list of each of the output (column) vectors you get when you apply it to each basis vectors as input. Multiplication is actually function composition: if A is the matrix for g and B is the matrix for f, then AB is the matrix for the function `x -> g(f(x))`.
skedaddle 4y ago
In the row picture, the elements of a matrix are the coefficients of linear functions acting on its domain. In the column picture they are coordinates in its image. Any matrix A or B can be interpreted from either point of view on its own. When you take their product AB, each of A's functions (row picture) is evaluated on each of B's points (column picture). This gives an associative (but not commutative) algebra. If you go around the column picture with an operator like A.B=AB^T, you get (A.B).C = (AB^T).C = AB^TC^T A.(B.C) = A.(BC^T) = A(BC^T)^T = ACB^T The two formulas are not equal, and second involves "traditional" matrix multiplication. You can compute products like this operationally though, as long as you work from left to right.

ecnahc515 4y ago

It's probably related to how we read and write. Top down, left to right. Rows are the entries, columns are attributes. Your list will potentially grow forever, whereas the attributes are typically fixed. Your width is typically fixed, but you can continue writing top down and continue on a new sheet/etc, as you add more rows. It just makes sense with how we typically organize information on pen/paper.

tptacek 4y ago

I'm bad at math, and know some linear algebra only because it's useful in cryptanalysis, but the way I keep this in my head is that matrix multiplication, mechanically, is just repeated application of matrix-vector multiplication; you're just doing multiple columns, not just one. And then, matrix-column vector multiplication is linear function application, right? And you can do column-times-row too, right? It's something like the sum of multiple outer products? (These question marks are all genuine! I probably have most of this wrong!)

theteapot 4y ago

It's a dark pattern to make make Matrix math seem more complicated than it is.

geokon 4y ago

The way things are you have a nice set of properties which allow you to manipulate system of matrix equations - but it's just notation and convention. You could do column by row as well - it'd be all the same math just effectively transposed. If you defined matrix multiplication as row times row then you are either limited to square matrices or you loose some of the convenient properties of matrix multiplication ABC = A(BC) = (AB)C (AB)^T = B^T A^T and most importantly AB =/= BA

ndriscoll 4y ago

Vectors are written as columns. If I have a linear function, I can describe it to you by just telling you what it does to each basis element, so that's what a matrix is: a list of each of the output (column) vectors you get when you apply it to each basis vectors as input. Multiplication is actually function composition: if A is the matrix for g and B is the matrix for f, then AB is the matrix for the function `x -> g(f(x))`.

skedaddle 4y ago

In the row picture, the elements of a matrix are the coefficients of linear functions acting on its domain. In the column picture they are coordinates in its image. Any matrix A or B can be interpreted from either point of view on its own. When you take their product AB, each of A's functions (row picture) is evaluated on each of B's points (column picture). This gives an associative (but not commutative) algebra. If you go around the column picture with an operator like A.B=AB^T, you get (A.B).C = (AB^T).C = AB^TC^T A.(B.C) = A.(BC^T) = A(BC^T)^T = ACB^T The two formulas are not equal, and second involves "traditional" matrix multiplication. You can compute products like this operationally though, as long as you work from left to right.