Linear-Algebra
Why are Determinants Like That? (YT)

Definition and Derivation

In an dimensional Vector Space (specifically ) we want to define a function to calculate the area/volume bounded by linearly independent vectors.

2D

First we will start in with basis vectors so some natural properties of this area function (call it ) are as following:

However we can notice that setting in 5 gives that but using the rule in 5 we get

Justifying why in 4 the domain is and not .

Next we try but applying rule 5 gives

The sign gives the orientation of the parallelogram. If then orientation is negative and if the orientation is positive. For convention we choose orientation to be positive when going from to is counter-clockwise. For example because going from to is a counter-clock wise rotation. However this is just convention you could define so but you must pick some orientation to be positive and convention is counter-clockwise.

Computation

By writing as linear combinations of the basis vectors we get and applying or rules we get

Now instead of the function taking multiple vectors as input we have the function call it where is an (in this case ) matrix with column vectors

Generalisation to

In we have basis vectors where is a column vector with all s except a in the th row. So our axioms for (no for volume) are as follows:

  1. - if any two vectors are the same they bound no volume. Holds if there are duplicates in any 2 slots.
  2. - a zero in any slot means the whole thing
  3. - works in any two slots. Combining 4 and 5 gives the property of multi-linearity - linear in every slot.

By computing as we have two of the same vectors but expanding out using 5 gives. This generalises to any slot meaning swapping any two vectors flips the sign again giving rise to the idea of orientation.

We can also see that the sign of corresponds to the number of swaps needed to reach the permutation more specifically the sign is negative for an odd amount of swaps and positive for an even amount. From this we define a function where is a permutation of and equivalently defined as

Computation

Leibniz Formula

We first write a vector in terms of the basis vectors giving

The reason why we get from to is because the first sum goes over every sequence with length where the elements are any number from - but as any volume with a repeated vector is zero we only consider the subset of those sequences where every element is unique which is the permutations of .

Now instead of the function taking multiple vectors as input we have the function call it where is an matrix with column vectors .

Recursive Formula

Using the Leibniz formula we can derive a recursive formula. Given an matrix

Proof Note that in this proof subscripts are flipped e.g. is the th element of the th column vector as we are emphasising column vectors but it doesn’t matter as the determinant is constant under transpose.

We will go through every permutation by going through every case where then up to . When considering the case ( has been removed from ) we will rename the remaining numbers/index to be from and rename the remaining vectors elements . So the remaining elements are denoted as .

For every even we pick up a minus sign formally . The reason why is because for any permutation keep in place then do all the required swaps to get the remaining elements in order then you can move to the right place by repeatedly swapping adjacent elements to move further in the sequence. If is odd this requires an even amount of swaps not affecting the sign but if is even this requires an odd amount of swaps flipping the sign.

Putting this together we get

The trick is that we can notice that each term has the same format as a determinant but with dimension and if we pay attention to the fact we have removed the first column and the th row we realise it is the determinant of the minor .