Matrix determinant from the exterior algebra viewpoint
In every linear algebra course matrix determinant is a must. Often it is introduced in the following form:
Definition 1 (Determinant) Let
The definition above has a lot of advantages, but it also has an important drawback — the “why” of this construction is hidden and appears only later in a long list of its properties.
We’ll take an alternative viewpoint, which I have learned from Darling (1994, chap. 1), and is based around the exterior algebra.
All our vector spaces will be finite-dimensional.
We will assume that all vector spaces considered are over real numbers
Motivational examples
Consider
- Bilinearity:
and . - Antisymmetry:
.
Geometrically we can think of it as of a signed area of the parallelepiped spanned by
For three vectors
- Trilinearity:
(and similarly in and arguments). - Antisymmetry: when we swap any two arguments the sign changes, e.g.,
.
Exterior algebra will be a generalisation of the above construction beyond the three-dimensional space
Exterior algebra
Let’s start with the natural definition:
Definition 2 (Antisymmetric multilinear function) Let
As we have seen above both
Note that for every
Exercise 1 Let
is antisymmetric, i.e., for every . is alternating, i.e., for every .
Generalise to multilinear mappings
Expand
Now we are ready to construct (a particular) exterior algebra.
Definition 3 (Second exterior power) Let
- The wedge
operator is bilinear, i.e., and . is antisymmetric, i.e., (or, equivalently, ).- If
is a basis of , then is a basis of .
Note that
We just need to prove that such a space actually exists (this construction can be skipped at the first reading): similarly to the tensor space, build the free vector space on the set
Then define
If we had introduced the determinant by other means, we could construct the exterior algebra
Analogously we can construct:
Definition 4 (Exterior power) Let
is a basis of
Exercise 2 Show that if
The introduced space can be used to convert between antisymmetric multilinear and linear functions by the means of the universal property:
Theorem 1 (Universal property) Let
Proof. (Can be skipped at the first reading.)
As
We can use antisymmetry to show that by “sorting out” the elements such that
Now the uniqueness is proven by observing that antisymmetry and multilinearity uniquely prescribe the values at the basis elements of
Its importance is the following: to show that a linear map
Determinants
Finally, we can define the determinant. Note that if
Definition 5 (Determinant) Let
As it is antisymmetric and multilinear, we know that it induces a unique linear mapping
Because
In other words, determinant measures the volume stretch of the parallelepiped spanned by the vectors after they are transformed by the mapping.
I like this geometric intuition, especially that it is clear that determinant depends only on the linear map, rather than a particular matrix representation — it is independent on the chosen basis.
We can now show a number of lemmata.
Proposition 1 If
Proof. Obvious from the definition! Similarly, it’s clear that
Proposition 2 For every two mappings
Proof. For every set of vectors we have
Proposition 3 (Only invertible matrices have non-zero determinants) A mapping is an isomorphism if and only if it has non-zero determinant.
Proof. If the mapping is invertible, then
Now assume that the mapping is non-invertible. This means that there exists a non-zero vector
Let’s now connect the usual definition of the determinant to the one coming from exterior algebra:
Proposition 4 (Recovering the standard expression) Let
Proof. Observe that
Now we see that repeated indices give zero contribution to this sum, so we can only consider the indices which are permutations of
Going just a bit further into exterior algebra we can also show that matrix transposition does not change the determinant.
To represent matrix transposition, we will use the dual mapping: if
We can therefore build the
We will formally show that
Proposition 5 (Determinant of the transpose) Let
Proof. To do this we will need an isomorphism
Having this, the proof becomes straightforward calculation:
Establishing such isomorphisms is quite a nice technique, which also can be used to prove
Proposition 6 (Determinant of a block-diagonal matrix) Let
Then
Proof. We will use this approach: there exists an isomorphism
Proposition 7 (Determinant of an upper-triangular matrix) Let
Once proven, this result can also be used for lower-triangular matrices due to Proposition 5.
Proof. Recall that whenever there is
Acknowledgements
I would like to thank Adam Klukowski for helpful editing suggestions.