Let F be a field. A matrix = M_{nxn}(F)

Is tr(A)=trace(A) a member of the feild F? (Does it have to be a member of the feild?)

Just to be sure you got it, ill explain it in another way:

When you calculate tr(A) (when summing the numbers) do you refer to the members a_{i,i }As normal numbers with the normal "+" function we all know, or add them like you add members of the feild (if in the feild 0=2 then 2+2=2)

Please answer, this is very important

Ehrlich
Mar 25, 2017

#1**+2 **

Did I fail you last time?!!!

I haven't checked, and it's a long time since I did this sort of thing, but:

1. I think it wouldn't make sense to use anything other than the field's own definition of addition in calculating the trace (sum of diagonal elements).

2. Hence the trace must be a member of the field.

If the answer is important then you should double check this with someone more qualified to answer!

Alan
Mar 25, 2017

#2**-1 **

Then ive misunderstood my professor.

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he said he can prove that if F is a field, Char(F)=0, A matrix =Â Mnxm(F), BÂ matrix =Â M_{mxn}(F),

Â

AB=I_{n}

Â

And

Â

BA=I_{m}

Â

Then A is Â aÂ square matrix.

Â

his way: at the beggining he proved that ifÂ A matrix =Â Mnxm(F), BÂ matrix =Â M_{mxn}(F) then tr(AB)=tr(BA).

Â

then he said n=tr(I_{n})=tr(AB)=tr(BA)=tr(I_{m})=m then n=m

Â

BUT HE DIDNT EVEN USE THE FACT CHAR(F)=0 AND HE SAID ITS HARD TO PROVE IT WITHOUT HAVING THE FACT CHAR(F)=0. I KNOW I MISUNDERSTOOD HIM BUT MY QUESTION IS WHERE. ITS FREAKING ME OUT.

Â

by the way, what are you doing nowadays? are you a professor or something?

Ehrlich
Mar 25, 2017

#3**-1 **

Then ive misunderstood my professor.

he can prove that if F is a field, Char(F)=0, A matrix = Â M_{nxm}(F), B matrix = Â M_{mxn}(F),

AB=In

And

BA=Im

Then A is Â aÂ square matrix.

his way: at the beggining he proved that ifÂ A matrix =Â M_{nxm}(F), BÂ matrix =Â M_{mxn}(F) then tr(AB)=tr(BA).

then he said n=tr(In)=tr(AB)=tr(BA)=tr(Im)=m then n=m

BUT HE DIDNT EVEN USE THE FACT CHAR(F)=0 AND HE SAID ITS HARD TO PROVE IT WITHOUT HAVING THE FACT CHAR(F)=0. I KNOW I MISUNDERSTOOD HIM BUT MY QUESTION IS WHERE. ITS FREAKING ME OUT.

by the way, what are you doing nowadays? are you a professor or something?

Â

Â

Â

sorry i resent it because something fucked up for some reason

Ehrlich
Mar 25, 2017

#4**0 **

I'm afraid my knowledge is insufficient here. Perhaps this will help:

http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/tracenorm.pdf

I am not and never have been a professor!

Alan
Mar 25, 2017

#5**-1 **

Then how do you know so much about things and stuff?

What university did you graduate from?

Ehrlich
Mar 25, 2017