It's hip to be square (or rectangular) - solving equations with matrices

equations, linear algebra

Arranging numbers in rectangles to do math turns out to often be more efficient than iteration.

Sometimes you have to solve a bunch of equations. Let’s take a simple example. Let’s say you’re converting a bunch of temperatures from Celsius to Fahrenheit.

You’ve got your list of $^oC$ temperatures: 140, 145, 160, 163, 170, 175…

And you know that:

$$ ^oF =\space ^oC \frac{9}{5} + 32 $$

If you’ve got more than 5 temperatures, using the calculator gets annoying. If you’re familiar with spreadsheets, this is when you’d create a formula in one of them. Linear algebra can be used very much like a spread sheet.

We can matrix multiplication to do our work quickly and efficiently.

When you multiply a matrix by a vector, you multiply each row in the matrix by each item in the vector and sum across the row.

For example:

$$ \begin{bmatrix} A & B \\ C & D \end{bmatrix} * \begin{bmatrix} E \\ F \end{bmatrix} = \begin{bmatrix} A E + B F \\ C E + D F \end{bmatrix} $$

So, to set this up with tempratures we have:

$$ \begin{bmatrix} 140 & 1 \\ 145 & 1 \\ 160 & 1 \\ 163 & 1 \\ 170 & 1 \\ 175 & 1 \\ \end{bmatrix} * \begin{bmatrix} \frac{9}{5} \\ 32 \end{bmatrix} $$

If you’re using Octave and import the temperatures in - or as in the example type them out - this is pretty easy to setup.

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% vector of temparutures
T = [140; 145; 160; 163; 170; 175]
 
% add the ones
T = [T ones(length(T), 1)]
 
c = [9/5; 32]

% do the math
F = T*c

F =

   284.00
   293.00
   320.00
   325.40
   338.00
   347.00

If you’re a programmer, you’re probably saying to yourself that you can just do this with iteration and a CtoF function. The interesting thing is that matrix operations like this are generally more efficient than iteration.

You can also use a similar concept to solve multiple unknowns in multiple equations.

For example if we have 4x+3y=17 and 10x - 4x = 8 we can represent those two equations like we did above.

$$ \begin{bmatrix} 4 & 3 \\ 10 & -4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 17 \\ 8 \end{bmatrix} $$

So, we’ve setup AX = B, to solve for X we multiply both sides by the inverse of A:

$$ A^{-1}AX=A^{-1}B $$

And $A^{-1}A$ is $I$, the identity matrix, so $IX=X$

$$ X=A^{-1}B $$

So, in Octave we have:

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pinv([4 3;10 -4])*[17; 8]
ans =

   2.0000
   3.0000

So x is 2 and y is 3. This equation set is simple enough that you can visually verify. The neat thing is that this works for much larger numbers of unkowns too…. and those are a real pain to do by hand!