/sci/ - Science & Math

No

>>4642549
bet you cant show a counterexample

>>4642556

>>4642561
o right, i thought we were talking about nXn matrixes

>>4642567
don't make assumptions, scrub

>>4642561
fuck

>>4642567
>pleb

>>4642561
>implying anyone ever actually uses nxm matrices
I'm a Physics PhD student and I never need to deal with that shit.

>>4642542
yes. as saying to say "inverses" implies we are working in a field, so we'll have nxn invertible matrices and zero only

>>4642594
Can you elaborate on this?

>>4642602
Group & Field theory, ideas of additive and multiplicative inverses. Matrices with different dimensionality are from separate groups/fields, and as such operations between them are invalid.

("Operation" means "you take on element from a group/field, combine it with another element from that same group/field, and the result is another element in that group/field")

>>4642594
>>4642611
The fuck am I reading.

>>4642624
You're reading someone trying to change the definitions to suit the point he wants to make.

>>4642591
>Physics PhD

hahaha what the fuck

>>4642626
Hang on, what is it that you're trying to say?
I'm >>4642611 , but I'm not >>4642594 .

>>4642591
I call bullshit, anon. Where are you studying? What field of research?

>>4642636
Apologies, I meant >>4642594 only.

>m x n matrices with m not equal no n

The fuck is that? reminds me of highschool when we were learning to multiply matrices. I have not seen a non square matrix ever since.

>m x n matrices with m not equal no n

The fuck is that? reminds me of highschool when we were learning to multiply matrices. I have not seen a non square matrix (except vector colums) ever since.

OP, for any two matrices, it isn't enough. Matrices can have left inverses and right inverses.

For a square matrices, though, it IS enough.

>>4642591
>>4642640
Are there actually any non-contrived applications for non-square matrices? (other than just row/column vectors)

>>4642594
No, it does not imply that at all. The word "inverse" in the context of the sentence "two matrices multiply together to make the identity matrix" has fuck all to do with group theory, since then you'd use "tensors" and oddly-named products of things. instead of "matrix" and "multiply."

Someone saying "multiply two matrices together," or words to that effect, is telling you at what level they are capable of holding discussion. If you insist that everyone on /sci/ look up the formal definition of every mathematical term they use, ever, fine, but don't expect many people to join you in your elegantly formal delusion-bubble.

(I don't know much group theory)

>>4642594
Lolwat? You should review your basic algebra.

>>4642611
You are forgetting about categories and groupoids, you freshman you. Spesifically the category of square matrices with (invertible) rectangular matrices between them. Shit forms a category (groupoid) using matrix multiplication as composition.

>>4642649
>calls fellow poster a 'freshman'
>can't spell "specifically"

>>4642644
Jacobians / when differentialting functions between spaces of different dimension (for example in differential topology/geometry).

>>4642660
This. Also non-square matrices arise in computation all the time.
>differentialting

I almost certain if we are talking about square matrix, if ab=1 then b=a^-1 you don't need to show ba=1 .
trying to prove it right now

>>4642643
*a pair of

There are a good amount of ways to prove it (which don't necessarily need the group theory other people are talking about). For example, if <span class="math">AB = I[/spoiler] and <span class="math">A, B[/spoiler] are square, then <span class="math">A[/spoiler] is one of <span class="math">B[/spoiler]'s left inverses and <span class="math">B[/spoiler] is <span class="math">A[/spoiler]'s right inverses.

It is known that a matrix <span class="math">B[/spoiler] is invertible iff its echelon form has pivots in every row and column. Since <span class="math">Bx = 0 \Rightarrow ABx = 0 \Rightarrow x = 0[/spoiler], then <span class="math">B[/spoiler] has linearly independent columns/vectors. Then, in echelon form, <span class="math">B[/spoiler] has pivots in every column (quick proof: if <span class="math">B[/spoiler] didn't have pivots in every column, it would have free variables, which negate its columns' linear independence). Since <span class="math">B[/spoiler] is <span class="math">n \times n[/spoiler] it has pivots in every row, too. Thus, B is invertible and A is its unique inverse.

Nonsquare matrixes have indeed applications, if for example conduct an experiment with a small set n of parameters, and make a large set m of measurements, you end up with nonsquare ( nxm ) matrices. If you conduct your experiment to find some parameters, this usually leads to a fitting problem, eg least-squares fit.
To be able to inverse non-square or singular matrices, the Moore–Penrose pseudoinverse was introduced.