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Interesting article... though to be honest, I'm not sure I buy the premise. In the article, he states: "Let's call the matrix of all (xi 1) `X` and let's call

by spott 3y ago

Interesting article... though to be honest, I'm not sure I buy the premise. In the article, he states: "Let's call the matrix of all (xi 1) `X` and let's call the vector of all yi `Y`", and then states that the units of `X` are (m 1). But if you instead say the units of `X` are just m, the "1" is in units "m", then the problem goes away, the whole matrix has the same units ("m"), and everything works fine. I'll be honest in that I've never thought of a matrix having multiple units per column... and I can't think of any example where you aren't just putting the units in the wrong place. Let me know if you have a concrete example that might work.

adastra22 3y ago
If you multiply by a transform where the first entry is 1, the second entry is d/dx, the third entry is d^2/dx^2, etc. You will get a different unit for each entry of the vector. Also most of economics consists of linear systems with non homogeneous units. The first entry could be bushels of wheat, the next demand for steel, etc.
- spott 3y ago
  One way to think of this is that there are always multiple ways to break up the units when moving a system of equations into matrix form. What I’m trying to say is that there is always one way to break up the units that leads to matrices and vectors with a consistent unit. Transform vectors like you mentioned can have an implicit 1 with the right units in them in order to make the vector have a consistent unit. > Also most of economics consists of linear systems with non homogeneous units. The system of equations needs to have some sort of unit consistency. Which means there is some way to split up the units to keep the matrices and vectors with consistent units or the math doesn’t actually make sense.
  - adastra22 3y ago
    I don't understand the point you are making. These systems are intrinsically non-homogeneous in their typing. In input-output analysis in economics, for example, the elements of a vector represent the amounts of a commodity (coal, steel, electricity, etc.), and the columns of the demand matrix represent how much of each commodity is required to produce one unit of that commodity as output. So the type of row 1 is "kg of coal", the type of row 2 is "kg of steel", the type of row 3 is "kJ of energy", etc. Given this matrix and a vector representing the starting quantities, you can do some linear algebra to get a matrix representing how much of each quantity to allocate to each sector, and a vector representing the resulting output. The type of these vectors are "[kg Coal, kg Steel, kJ electricity, ...]" I don't now how you can "have an implicit 1 with the right units in them in order to make the vector have a consistent unit" given this setup.
    - kragen 3y ago
      being implicitly non-homogeneous doesn't mean they're logically intractable you can still imagine some kind of static analysis that gives you an error if you accidentally write an algorithm that might add kgs of steel to kilojoules, and which is sufficiently powerful to accept at least the commonly-used numerical algorithms; not just matrix multiply but, in your example, probably the simplex algorithm (we could call that analysis a 'type system', but perhaps obviously, it is significantly different from the type systems we're most familiar with) i agree that this doesn't end up with vectors of homogeneous types, but it does end up with vectors of the correct type, which might be what was meant by 'consistent'
      - adastra22 3y ago
        Yes, ok I agree with this.
  - kragen 3y ago
    > there is always one way to break up the units that leads to matrices and vectors with a consistent unit. maybe i don't understand what you're saying but it sounds like you don't understand the problem it's not that the units are inconsistent in the sense of adding kilograms to kilojoules; they're just inhomogeneous you have a potentially large matrix in which potentially every cell has different units. so type systems that require the matrix cells to all be of the same type aren't helpful unless that type is something machine-oriented like `real*8` moreover this is intrinsic to the problem, or at least the decision to use linear algebra on the problem an additional difficulty for things like *gemm or lu factorization is that you want them to be applicable to matrices of any size with any type of units of measurement, as long as they're consistent; so they're parametrically polymorphic over units of measurement
kragen 3y ago
i hadn't thought of it either before auditing an introductory numerical methods class, but in retrospect the page i linked does explain this, repeatedly, in the comments; i just didn't understand it at the time basically it's very very common

by spott 3y ago

adastra22 3y ago

If you multiply by a transform where the first entry is 1, the second entry is d/dx, the third entry is d^2/dx^2, etc. You will get a different unit for each entry of the vector. Also most of economics consists of linear systems with non homogeneous units. The first entry could be bushels of wheat, the next demand for steel, etc.

spott 3y ago
One way to think of this is that there are always multiple ways to break up the units when moving a system of equations into matrix form. What I’m trying to say is that there is always one way to break up the units that leads to matrices and vectors with a consistent unit. Transform vectors like you mentioned can have an implicit 1 with the right units in them in order to make the vector have a consistent unit. > Also most of economics consists of linear systems with non homogeneous units. The system of equations needs to have some sort of unit consistency. Which means there is some way to split up the units to keep the matrices and vectors with consistent units or the math doesn’t actually make sense.
- adastra22 3y ago
  I don't understand the point you are making. These systems are intrinsically non-homogeneous in their typing. In input-output analysis in economics, for example, the elements of a vector represent the amounts of a commodity (coal, steel, electricity, etc.), and the columns of the demand matrix represent how much of each commodity is required to produce one unit of that commodity as output. So the type of row 1 is "kg of coal", the type of row 2 is "kg of steel", the type of row 3 is "kJ of energy", etc. Given this matrix and a vector representing the starting quantities, you can do some linear algebra to get a matrix representing how much of each quantity to allocate to each sector, and a vector representing the resulting output. The type of these vectors are "[kg Coal, kg Steel, kJ electricity, ...]" I don't now how you can "have an implicit 1 with the right units in them in order to make the vector have a consistent unit" given this setup.
  - kragen 3y ago
    being implicitly non-homogeneous doesn't mean they're logically intractable you can still imagine some kind of static analysis that gives you an error if you accidentally write an algorithm that might add kgs of steel to kilojoules, and which is sufficiently powerful to accept at least the commonly-used numerical algorithms; not just matrix multiply but, in your example, probably the simplex algorithm (we could call that analysis a 'type system', but perhaps obviously, it is significantly different from the type systems we're most familiar with) i agree that this doesn't end up with vectors of homogeneous types, but it does end up with vectors of the correct type, which might be what was meant by 'consistent'
    - adastra22 3y ago
      Yes, ok I agree with this.
- kragen 3y ago
  > there is always one way to break up the units that leads to matrices and vectors with a consistent unit. maybe i don't understand what you're saying but it sounds like you don't understand the problem it's not that the units are inconsistent in the sense of adding kilograms to kilojoules; they're just inhomogeneous you have a potentially large matrix in which potentially every cell has different units. so type systems that require the matrix cells to all be of the same type aren't helpful unless that type is something machine-oriented like `real*8` moreover this is intrinsic to the problem, or at least the decision to use linear algebra on the problem an additional difficulty for things like *gemm or lu factorization is that you want them to be applicable to matrices of any size with any type of units of measurement, as long as they're consistent; so they're parametrically polymorphic over units of measurement

kragen 3y ago

i hadn't thought of it either before auditing an introductory numerical methods class, but in retrospect the page i linked does explain this, repeatedly, in the comments; i just didn't understand it at the time basically it's very very common