I always liked the interpretation of the determinant as measuring the expansion/contraction of n-dimensional volumes induced by a linear map, with the sign being negative if the orientation of space is flipped. This makes various properties intuitively clear such as non-zero determinant being equivalent to invertibility.
Yup, determinant is how much the volume stretches. And trace is how much the vectors stay pointing in the same direction (average dot product of v and Av). This explains why trace of 90 degree rotation in 2D space is zero, why trace of projection onto a subspace is the dimension of that subspace, and so on.
My intuition for exp is that it tells you how an infinitesimal change accumulates over finite time (think compound interest). So the above expression is equivalent to det(I+εA)=1+εtr(A)+O(ε2). Thus we should think ‘If I perturb the identity matrix, then the amount by which the unit cube grows is proportional to the extent to which each vector is being stretched in the direction it was already pointing’.
Hmm, this seems wrong but fixable. Namely, exp(A) is close to (I+A/n)^n, so raising both sides of det(exp(A))=exp(tr(A)) to the power of 1/n gives something like what we want. Still a bit too algebraic though, I wonder if we can do better.
I think the determinant is more mathematically fundamental than the concept of volume. It just seems the other way around because we use volumes in every day life.
I think the good abstract way to think about the determinant is in terms of induced maps on the top exterior power. If you have an n dimensional vector space V and an endomorphism L:V→V, this induces a map ∧nV→∧nV, and since ∧nV is always one-dimensional this map must be of the form v→kv for some scalar k in the ground field. It’s this k that is the determinant of L.
This is indeed more fundamental than the concept of volume. We can interpret exterior powers as corresponding to volume if we’re working over a local field, for example, but actually the concept of exterior power generalizes far beyond this special case. This is why the determinant still preserves its nice properties even if we work over an arbitrary commtuative ring; since such rings still have exterior powers behaving in the usual way.
I didn’t present it like this in this post because it’s actually not too easy to introduce the concept of “exterior power” without the post becoming too abstract.
This is close to one thing I’ve been thinking about myself. The determinant is well defined for endomorphisms on finitely-generated projective modules over any ring. But the ‘top exterior power’ definition doesn’t work there because such things do not have a dimension. There are two ways I’ve seen for nevertheless defining the determinant.
View the module as a sheaf of modules over the spectrum of the ring. Then the dimension is constant on each connected component, so you can take the top exterior power on each and then glue them back together.
Use the fact that finitely-generated projective modules are precisely those which are the direct summands of a free module. So given an endomorphism A:V→V you can write V⊕V′=Rn and then define det(A)=det(A⊕idV′).
These both give the same answer. However, I don’t like the first definition because it feels very piecemeal and nonuniform, and I don’t like the second because it is effectively picking a basis. So I’ve been working on by own definition where instead of defining Λn(V) for natural numbers n we instead define ΛW(V) for finitely-generated projective modules W. Then the determinant is defined via ΛV(V).
I’m curious about this. I can see a reasonable way to define ΛW(V) in terms of sheaves of modules over Spec(R): Over each connected component, W has some constant dimension n, so we just let ΛW(V) be Λn(V) over that component. But it sounds like you might not like this definition, and I’d be interested to know if you had a better way of defining ΛW(V) (which will probably end up being equivalent to this). [Edit: Perhaps something in terms of generators and relations, with the generators being linear maps W→V?]
I’m curious about this. I can see a reasonable way to define ΛW(V) in terms of sheaves of modules over Spec(R): Over each connected component, W has some constant dimension n, so we just let ΛW(V) be Λn(V) over that component.
If we call this construction F(W,V) then the construction I’m thinking of is F(W,V)⊗F(W,W)∗. Note that F(W,W) is locally 1-dimensional, so my construction is locally isomorphic to yours but globally twisted. It depends on W via more than just its local dimension. Also note that with this definition we will get that ΛV(V) is always isomorphic to R.
But it sounds like you might not like this definition,
Right. I’m hoping for a simple definition that captures what the determinant ‘really means’ in the most general case. So it would be nice if it could be defined with just linear algebra without having to bring in the machinery of the spectrum.
and I’d be interested to know if you had a better way of defining ΛW(V) (which will probably end up being equivalent to this).
I’m still looking for a nice definition. Here’s what I’ve got so far.
If we pick a basis of Rn then it induces a bijection between Hom(Rn,V) and V×⋯×V. So we could define a map Hom(Rn,V)→U to be ‘alternating’ if and only if the corresponding map V×⋯×V→U is alternating. The interesting thing I noticed about this definition is that it doesn’t depend on which basis you pick for Rn. So I have some hope that since this construction isn’t basis dependent, I might be able to write down a basis-independent definition of it. Then it would apply equally well with Rn replaced with W, whereupon we can define ΛW(V) as the universal alternating map out of Hom(W,V).
[Edit: Perhaps something in terms of generators and relations, with the generators being linear maps W→V?]
Yeah exactly. That’s probably a simpler way to say what I was describing above. One embarrassing thing is that I don’t even know how to describe the simplest relations, i.e. what the0s should be.
If we call this construction F(W,V) then the construction I’m thinking of is F(W,V)⊗F(W,W)∗. Note that F(W,W) is locally 1-dimensional, so my construction is locally isomorphic to yours but globally twisted. It depends on W via more than just its local dimension. Also note that with this definition we will get that ΛV(V) is always isomorphic to R
Oh right, I was picturing W being free on connected components when I suggested that. Silly me.
If we pick a basis of Rn then it induces a bijection between Hom(Rn,V) and V×⋯×V. So we could define a map Hom(Rn,V)→U to be ‘alternating’ if and only if the corresponding map V×⋯×V→U is alternating. The interesting thing I noticed about this definition is that it doesn’t depend on which basis you pick for Rn. So I have some hope that since this construction isn’t basis dependent, I might be able to write down a basis-independent definition of it.
F is alternating if F(f∘g)=det(g)F(f), right? So if we’re willing to accept kludgy definitions of determinant in the process of defining ΛW(V), then we’re all set, and if not, then we’ll essentially need another way to define determinant for projective modules because that’s equivalent to defining an alternating map?
if not, then we’ll essentially need another way to define determinant for projective modules because that’s equivalent to defining an alternating map?
There’s a lot of cases in mathematics where two notions can be stated in terms of each other, but it doesn’t tell us which order to define things in.
The only other thought I have is that I have to use the fact that W is projective and finitely generated. This is equivalent to W being dualisable. So the definition is likely to use W∗ somewhere.
For what it’s worth, I strongly agree (1) that exterior algebra is the Right Way to think about determinants, conditional on there being other reasons in your life for knowing exterior algebra, and (2) that doing it that way in this post would have had too much overhead.
I felt dumb recently when I noticed that the determinant is sort of “the absolute value for matrices”, considering that it’s literally written using the same signs as the absolute value. Although I guess the determinant of the representation of a complex number as a matrix is ∣∣∣a−bba∣∣∣=a2+b2=|a+bi|2, not |a+bi|. The “signed volume” idea seems related to this, insofar as multiplying a complex number z1 by another z2 will stretch / smush z1 by |z2| (in addition to rotating it).
This is actually (hopefully) the first post in a series, and I’ll talk about this way of looking at the determinant in a subsequent post. It actually generalizes past vector spaces over R if you do it in the appropriate way.
The problem is that the equivalence of this to the usual determinant is not easy to prove unless you have some machinery at your disposal already. It’s “obvious” that volume should be invariant under skew translations, for example, but the easiest proof I know of this simply goes through the determinant of a skew translation matrix and shows it’s equal to 1.
If you have the time, try thinking of how you would prove that the characterization you give here is actually equivalent to the characterization of the determinant in the post—alternating and multilinear map that’s equal to 1 on the identity matrix. The “multilinear” part turns out to be rather tricky to establish properly.
Thinking about how to prove the multilinearity of the volume of a parallelepiped definition I like this sketched approach:
The two dimensional case is a “cute” problem involving rearranging triangles and ordinary areas (or you solve this case in any other way you want). The general case then follows from linearity of integrals (you get the higher dimensional cases by integrating the two dimensional case appropriately).
So, this is not exactly a rigorous proof, but off the top of my head, I would justify/remember the properties like: identity has determinant 1 because it doesn’t change the size of any volumes, determinant is alternating because swapping two axes of a parallelpiped is like reflecting those axes in a mirror, changing the orientation. Multilinearity is equivalent to showing that the volume of a parallelpiped A=a1∧a2∧(...)∧an is linear in all the ais. But this follows since the volume of A is equal to the volume of a2∧(...)∧an multiplied by the component of a1 projected onto the axis orthogonal to a2,...,an, which is clearly linear in a1. This last fact is a bit weird, to justify intuitively I imagine having a straight tower of blocks which you then ‘slant’ by pulling the top in a given direction without changing the volume, this corresponding to the components of a1 not orthogonal to a2,...,an.
I always liked the interpretation of the determinant as measuring the expansion/contraction of n-dimensional volumes induced by a linear map, with the sign being negative if the orientation of space is flipped. This makes various properties intuitively clear such as non-zero determinant being equivalent to invertibility.
Yup, determinant is how much the volume stretches. And trace is how much the vectors stay pointing in the same direction (average dot product of v and Av). This explains why trace of 90 degree rotation in 2D space is zero, why trace of projection onto a subspace is the dimension of that subspace, and so on.
Thank you for that intuition into the trace! That also helps make sense of det(exp(A))=exp(tr(A)).
Interesting, can you give a simple geometric explanation?
My intuition for exp is that it tells you how an infinitesimal change accumulates over finite time (think compound interest). So the above expression is equivalent to det(I+εA)=1+εtr(A)+O(ε2). Thus we should think ‘If I perturb the identity matrix, then the amount by which the unit cube grows is proportional to the extent to which each vector is being stretched in the direction it was already pointing’.
Hmm, this seems wrong but fixable. Namely, exp(A) is close to (I+A/n)^n, so raising both sides of det(exp(A))=exp(tr(A)) to the power of 1/n gives something like what we want. Still a bit too algebraic though, I wonder if we can do better.
Another thing to say is if A(0)=I then
I think the determinant is more mathematically fundamental than the concept of volume. It just seems the other way around because we use volumes in every day life.
I think the good abstract way to think about the determinant is in terms of induced maps on the top exterior power. If you have an n dimensional vector space V and an endomorphism L:V→V, this induces a map ∧nV→∧nV, and since ∧nV is always one-dimensional this map must be of the form v→kv for some scalar k in the ground field. It’s this k that is the determinant of L.
This is indeed more fundamental than the concept of volume. We can interpret exterior powers as corresponding to volume if we’re working over a local field, for example, but actually the concept of exterior power generalizes far beyond this special case. This is why the determinant still preserves its nice properties even if we work over an arbitrary commtuative ring; since such rings still have exterior powers behaving in the usual way.
I didn’t present it like this in this post because it’s actually not too easy to introduce the concept of “exterior power” without the post becoming too abstract.
This is close to one thing I’ve been thinking about myself. The determinant is well defined for endomorphisms on finitely-generated projective modules over any ring. But the ‘top exterior power’ definition doesn’t work there because such things do not have a dimension. There are two ways I’ve seen for nevertheless defining the determinant.
View the module as a sheaf of modules over the spectrum of the ring. Then the dimension is constant on each connected component, so you can take the top exterior power on each and then glue them back together.
Use the fact that finitely-generated projective modules are precisely those which are the direct summands of a free module. So given an endomorphism A:V→V you can write V⊕V′=Rn and then define det(A)=det(A⊕idV′).
These both give the same answer. However, I don’t like the first definition because it feels very piecemeal and nonuniform, and I don’t like the second because it is effectively picking a basis. So I’ve been working on by own definition where instead of defining Λn(V) for natural numbers n we instead define ΛW(V) for finitely-generated projective modules W. Then the determinant is defined via ΛV(V).
I’m curious about this. I can see a reasonable way to define ΛW(V) in terms of sheaves of modules over Spec(R): Over each connected component, W has some constant dimension n, so we just let ΛW(V) be Λn(V) over that component. But it sounds like you might not like this definition, and I’d be interested to know if you had a better way of defining ΛW(V) (which will probably end up being equivalent to this). [Edit: Perhaps something in terms of generators and relations, with the generators being linear maps W→V?]
If we call this construction F(W,V) then the construction I’m thinking of is F(W,V)⊗F(W,W)∗. Note that F(W,W) is locally 1-dimensional, so my construction is locally isomorphic to yours but globally twisted. It depends on W via more than just its local dimension. Also note that with this definition we will get that ΛV(V) is always isomorphic to R.
Right. I’m hoping for a simple definition that captures what the determinant ‘really means’ in the most general case. So it would be nice if it could be defined with just linear algebra without having to bring in the machinery of the spectrum.
I’m still looking for a nice definition. Here’s what I’ve got so far.
If we pick a basis of Rn then it induces a bijection between Hom(Rn,V) and V×⋯×V. So we could define a map Hom(Rn,V)→U to be ‘alternating’ if and only if the corresponding map V×⋯×V→U is alternating. The interesting thing I noticed about this definition is that it doesn’t depend on which basis you pick for Rn. So I have some hope that since this construction isn’t basis dependent, I might be able to write down a basis-independent definition of it. Then it would apply equally well with Rn replaced with W, whereupon we can define ΛW(V) as the universal alternating map out of Hom(W,V).
Yeah exactly. That’s probably a simpler way to say what I was describing above. One embarrassing thing is that I don’t even know how to describe the simplest relations, i.e. what the 0s should be.
Oh right, I was picturing W being free on connected components when I suggested that. Silly me.
F is alternating if F(f∘g)=det(g)F(f), right? So if we’re willing to accept kludgy definitions of determinant in the process of defining ΛW(V), then we’re all set, and if not, then we’ll essentially need another way to define determinant for projective modules because that’s equivalent to defining an alternating map?
There’s a lot of cases in mathematics where two notions can be stated in terms of each other, but it doesn’t tell us which order to define things in.
The only other thought I have is that I have to use the fact that W is projective and finitely generated. This is equivalent to W being dualisable. So the definition is likely to use W∗ somewhere.
For what it’s worth, I strongly agree (1) that exterior algebra is the Right Way to think about determinants, conditional on there being other reasons in your life for knowing exterior algebra, and (2) that doing it that way in this post would have had too much overhead.
(+1, came here to say this. Seems deficient to think of determinants without including this interpretation.)
I felt dumb recently when I noticed that the determinant is sort of “the absolute value for matrices”, considering that it’s literally written using the same signs as the absolute value. Although I guess the determinant of the representation of a complex number as a matrix is ∣∣∣a−bba∣∣∣=a2+b2=|a+bi|2, not |a+bi|. The “signed volume” idea seems related to this, insofar as multiplying a complex number z1 by another z2 will stretch / smush z1 by |z2| (in addition to rotating it).
This is actually (hopefully) the first post in a series, and I’ll talk about this way of looking at the determinant in a subsequent post. It actually generalizes past vector spaces over R if you do it in the appropriate way.
The problem is that the equivalence of this to the usual determinant is not easy to prove unless you have some machinery at your disposal already. It’s “obvious” that volume should be invariant under skew translations, for example, but the easiest proof I know of this simply goes through the determinant of a skew translation matrix and shows it’s equal to 1.
If you have the time, try thinking of how you would prove that the characterization you give here is actually equivalent to the characterization of the determinant in the post—alternating and multilinear map that’s equal to 1 on the identity matrix. The “multilinear” part turns out to be rather tricky to establish properly.
Thinking about how to prove the multilinearity of the volume of a parallelepiped definition I like this sketched approach:
The two dimensional case is a “cute” problem involving rearranging triangles and ordinary areas (or you solve this case in any other way you want). The general case then follows from linearity of integrals (you get the higher dimensional cases by integrating the two dimensional case appropriately).
So, this is not exactly a rigorous proof, but off the top of my head, I would justify/remember the properties like: identity has determinant 1 because it doesn’t change the size of any volumes, determinant is alternating because swapping two axes of a parallelpiped is like reflecting those axes in a mirror, changing the orientation. Multilinearity is equivalent to showing that the volume of a parallelpiped A=a1∧a2∧(...)∧an is linear in all the ais. But this follows since the volume of A is equal to the volume of a2∧(...)∧an multiplied by the component of a1 projected onto the axis orthogonal to a2,...,an, which is clearly linear in a1. This last fact is a bit weird, to justify intuitively I imagine having a straight tower of blocks which you then ‘slant’ by pulling the top in a given direction without changing the volume, this corresponding to the components of a1 not orthogonal to a2,...,an.
Yeah, that’s what I mean by saying it’s “obvious”. Similar to the change of variables theorem in that way.