Zulip Chat Archive

Stream: maths

Topic: matrix inverse


view this post on Zulip Chris Hughes (Jul 05 2019 at 14:19):

Is there a total matrix inverse function, defined on all matrices over a field that has the property that it returns left inverses when they exist, and right inverses when they exist, and (AB)1=B1A1(AB)^{-1} = B^{-1}A^{-1} for all matrices?

view this post on Zulip Kevin Buzzard (Jul 05 2019 at 14:20):

all matrices of any size? Or some fixed square size?

view this post on Zulip Chris Hughes (Jul 05 2019 at 14:20):

Any size

view this post on Zulip Kevin Buzzard (Jul 05 2019 at 14:21):

I can't even make this question typecheck

view this post on Zulip Kevin Buzzard (Jul 05 2019 at 14:22):

If A is an a * b matrix, what size is A^{-1} supposed to be?

view this post on Zulip Kevin Buzzard (Jul 05 2019 at 14:22):

b * a

view this post on Zulip Kevin Buzzard (Jul 05 2019 at 14:22):

and whether or not it's a left inverse or right inverse depends on whether a<b or b>a?

view this post on Zulip Chris Hughes (Jul 05 2019 at 14:23):

b * a

Yes.

view this post on Zulip Chris Hughes (Jul 05 2019 at 14:24):

and whether or not it's a left inverse or right inverse depends on whether a<b or b>a?

Yes, but maybe neither inverse exists.

view this post on Zulip Johan Commelin (Jul 05 2019 at 14:25):

Does this work: https://en.wikipedia.org/wiki/Invertible_matrix#In_relation_to_its_adjugate

view this post on Zulip Chris Hughes (Jul 05 2019 at 14:26):

That's only for square matrices.

view this post on Zulip Kevin Buzzard (Jul 05 2019 at 14:26):

The problem is with the non-square matrices.

view this post on Zulip Alexander Bentkamp (Jul 05 2019 at 14:26):

How about this: https://en.wikipedia.org/wiki/Generalized_inverse

view this post on Zulip Kevin Buzzard (Jul 05 2019 at 14:28):

I am concerned that they talk about the real numbers here. Actually @Abhimanyu Pallavi Sudhir has some paper about inverse matrices in more generality than is usually used by mathematicians, but he definitely needed the ground field to be the reals.

view this post on Zulip Chris Hughes (Jul 05 2019 at 14:28):

I need it for rationals.

view this post on Zulip Johan Commelin (Jul 05 2019 at 14:29):

I've never really thought about this... can't the adjugate be generalised? What if you take cofactors where you don't delete a row and a column but "at least one row, and at least one column" so that you end up with squares of size k-1 × k-1 where k = min (m,n).

view this post on Zulip Kevin Buzzard (Jul 05 2019 at 14:30):

Will that give you a left inverse where one exists?

view this post on Zulip Johan Commelin (Jul 05 2019 at 14:30):

I have no idea if that would work, but it gives the correct result for square matrices.

view this post on Zulip Kevin Buzzard (Jul 05 2019 at 14:30):

Wait -- where are you putting these cofactors?

view this post on Zulip Kevin Buzzard (Jul 05 2019 at 14:30):

They need to be arranged in a nice box

view this post on Zulip Johan Commelin (Jul 05 2019 at 14:30):

Aah, good point

view this post on Zulip Mario Carneiro (Jul 05 2019 at 14:31):

I think the pseudoinverse is what you want

view this post on Zulip Johan Commelin (Jul 05 2019 at 14:31):

/me goes back to commutative algebra

view this post on Zulip Kevin Buzzard (Jul 05 2019 at 15:08):

/me decides that if Mario is wrong then Chris will just ask again

view this post on Zulip Mario Carneiro (Jul 05 2019 at 15:13):

the moore-penrose inverse looks right, but also looks complicated -> https://en.wikipedia.org/wiki/Proofs_involving_the_Moore%E2%80%93Penrose_inverse

view this post on Zulip Mario Carneiro (Jul 05 2019 at 15:14):

it also needs singular value decomposition...

view this post on Zulip Mario Carneiro (Jul 05 2019 at 15:16):

and apparently both the SVD and the M-P inverse require the base field to be R or C

view this post on Zulip Johan Commelin (Jul 05 2019 at 15:17):

That's pretty ugly

view this post on Zulip Johan Commelin (Jul 05 2019 at 15:17):

How can such a thing work for R and C but not for general fields?

view this post on Zulip Mario Carneiro (Jul 05 2019 at 15:18):

I highly suspect that's not actually true, but you know mathematicians

view this post on Zulip Johan Commelin (Jul 05 2019 at 15:18):

I understand that certain things only work for alg.closed fields, or only for ordered fields...

view this post on Zulip Mario Carneiro (Jul 05 2019 at 15:18):

There is quite a lot that is done on fields that are R or C

view this post on Zulip Kevin Buzzard (Jul 05 2019 at 15:18):

I think that some of these definitions are quite analysis-y

view this post on Zulip Mario Carneiro (Jul 05 2019 at 15:18):

hilbert spaces and banach spaces come to mind

view this post on Zulip Kevin Buzzard (Jul 05 2019 at 15:18):

Abhi told me that his matrix inverse stuff didn't work over a general field.

view this post on Zulip Kevin Buzzard (Jul 05 2019 at 15:18):

hilbert spaces and banach spaces come to mind

That's analysis

view this post on Zulip Johan Commelin (Jul 05 2019 at 15:19):

Hmm..., maybe it is because they want to mention A^*?

view this post on Zulip Mario Carneiro (Jul 05 2019 at 15:19):

They are analysis because it says "R or C"

view this post on Zulip Johan Commelin (Jul 05 2019 at 15:19):

Also, the section "Definition" on wiki, is really just a theorem (-;

view this post on Zulip Mario Carneiro (Jul 05 2019 at 15:20):

The definition of a pesudoinverse, as a predicate, isn't hard

view this post on Zulip Mario Carneiro (Jul 05 2019 at 15:20):

but proving existence is hard and requires these analysis-y assumptions

view this post on Zulip Mario Carneiro (Jul 05 2019 at 15:21):

By the way @Chris Hughes these certainly don't necessarily exist over Q

view this post on Zulip Johan Commelin (Jul 05 2019 at 15:23):

I expect that you can do similar things over CM fields and totally real fields

view this post on Zulip Mario Carneiro (Jul 05 2019 at 15:23):

wait, now I'm not sure if that's true. It's not clear to me if eigenvalue decomposition of some kind is required

view this post on Zulip Kenny Lau (Jul 05 2019 at 15:23):

but Q is totally real

view this post on Zulip Johan Commelin (Jul 05 2019 at 15:25):

It sure is...

view this post on Zulip Johan Commelin (Jul 05 2019 at 15:25):

So it works over the algebraic reals

view this post on Zulip Johan Commelin (Jul 05 2019 at 15:25):

That existence claim is a first order formula, right?

view this post on Zulip Johan Commelin (Jul 05 2019 at 15:25):

Now I hope that you can do some Galois descent to go to subfields

view this post on Zulip Mario Carneiro (Jul 05 2019 at 15:26):

The inverse of an invertible matrix exists without any eigenvalue decomp

view this post on Zulip Mario Carneiro (Jul 05 2019 at 15:26):

So maybe the pseudoinverse also exists

view this post on Zulip Mario Carneiro (Jul 05 2019 at 15:26):

Doing SVD definitely requires eigenvalues though

view this post on Zulip Mario Carneiro (Jul 05 2019 at 15:27):

so that restricts application to algebraically closed fields and maybe real closed fields

view this post on Zulip Johan Commelin (Jul 05 2019 at 15:28):

I would like yo deduce after the fact that it is defined over subfields

view this post on Zulip Kevin Buzzard (Jul 05 2019 at 15:28):

I would like yo deduce after the fact that it is defined over subfields

You've been doing too many Shimura varieties.

view this post on Zulip Kevin Buzzard (Jul 05 2019 at 15:29):

I don't think it always works out so easily :-)

view this post on Zulip Johan Commelin (Jul 05 2019 at 15:31):

@Kevin Buzzard Look at https://en.wikipedia.org/wiki/Proofs_involving_the_Moore%E2%80%93Penrose_inverse#Definition

view this post on Zulip Johan Commelin (Jul 05 2019 at 15:32):

That characterisation is pretty algebraic..., so I would hope that it lends itself to Galois descent.

view this post on Zulip Kevin Buzzard (Jul 05 2019 at 15:32):

remember we call it Galois lift here

view this post on Zulip Johan Commelin (Jul 05 2019 at 15:33):

@Chris Hughes Do you want something that computes?

view this post on Zulip Johan Commelin (Jul 05 2019 at 15:33):

Because this seems extremely slow...

view this post on Zulip Abhimanyu Pallavi Sudhir (Jul 05 2019 at 15:42):

I am concerned that they talk about the real numbers here. Actually Abhimanyu Pallavi Sudhir has some paper about inverse matrices in more generality than is usually used by mathematicians, but he definitely needed the ground field to be the reals.

That's determinants, not inverses. And yeah, it's related to the SVD.

view this post on Zulip Kevin Buzzard (Jul 05 2019 at 15:43):

well, I guess they're related here. So you can take the determinant of a 2x3 matrix with real coefficients, right?

view this post on Zulip Kevin Buzzard (Jul 05 2019 at 15:43):

But if that 2x3 matrix had rational coefficients, would the determinant be rational?

view this post on Zulip Abhimanyu Pallavi Sudhir (Jul 05 2019 at 15:43):

No.

view this post on Zulip Abhimanyu Pallavi Sudhir (Jul 05 2019 at 15:43):

|det(1, 2)| = sqrt(5)

view this post on Zulip Abhimanyu Pallavi Sudhir (Jul 05 2019 at 15:44):

Or whatever 2^2+1 is.

view this post on Zulip Kevin Buzzard (Jul 05 2019 at 15:44):

it'll depend on what field you're in

view this post on Zulip Abhimanyu Pallavi Sudhir (Jul 05 2019 at 15:44):

Oh wait -- but it can be a vector of rational entries.

view this post on Zulip Abhimanyu Pallavi Sudhir (Jul 05 2019 at 15:44):

If you use the determinant in (Pyle, 1962).

view this post on Zulip Abhimanyu Pallavi Sudhir (Jul 05 2019 at 15:45):

Or an m-vector with rational entries.

view this post on Zulip Kevin Buzzard (Jul 05 2019 at 15:46):

what is m?

view this post on Zulip Abhimanyu Pallavi Sudhir (Jul 05 2019 at 15:46):

The dimension of the domain space.

view this post on Zulip Abhimanyu Pallavi Sudhir (Jul 05 2019 at 15:46):

So for n by m matrices the "determinant" can just be the wedge product of the m columns.

view this post on Zulip Abhimanyu Pallavi Sudhir (Jul 05 2019 at 15:47):

The Pyle determinant just flattens it out into an nCm dimensional vector.

view this post on Zulip Abhimanyu Pallavi Sudhir (Jul 05 2019 at 15:51):

and apparently both the SVD and the M-P inverse require the base field to be R or C

Presumably because the SVD involves the notion of scaling and rotations. If you only allowed scaling, you would need algebraic closure. But allowing rotations allows the complex behavior to be mimicked by real numbers.

view this post on Zulip Kevin Buzzard (Jul 08 2019 at 17:17):

@Chris Hughes were any of these attempts useful? This seems like a reasonable MO question to me.

view this post on Zulip Chris Hughes (Jul 08 2019 at 17:45):

I know that the answer is no now. None of the pseidoinverses are particularly useful for what I'm doing.


Last updated: May 06 2021 at 18:20 UTC