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        riddles >> putnam exam (pure math) >> matrix
        
(Message started by: temporary on Apr 8th, 2008, 9:24pm)
Title: matrix
Post by temporary on Apr 8th, 2008, 9:24pm
Construct a matrix where all numbers in it add up to 0, but with no negatives, no imaginary part greater than 0, and no 0s. This is not impossible, nor does it use infinity or something like that. Hint:[hide]there are infinite solutions[/hide]
Title: Re: matrix
Post by Obob on Apr 9th, 2008, 9:50am
Yeah, no...

You are asking for a series of positive numbers summing to zero.  Even in nonstandard interpretations of "number," this is not possible, because those interpretations do not also have an ordering.
Title: Re: matrix
Post by Michael_Dagg on Apr 9th, 2008, 12:55pm
[ 2 1 ]
[ 1 2 ]  

mod 3  works.
Title: Re: matrix
Post by Michael_Dagg on Apr 9th, 2008, 1:19pm
Since you might have also meant the sums along
the diagnoals then


[3 1 2]
[1 2 3]
[2 3 1]

mod 6  works.

Anyone got a general form?
Title: Re: matrix
Post by Obob on Apr 9th, 2008, 2:47pm
But is 2 > 0 mod 3?
Title: Re: matrix
Post by temporary on Apr 11th, 2008, 11:18pm
I have no idea what all this "mod" stuff is, but it is not part of the answer.
Title: Re: matrix
Post by Obob on Apr 11th, 2008, 11:25pm
Is it not part of the answer because there is no mod stuff in the answer, or is it not part of the answer because there is no answer?

I suspect both are true...
Title: Re: matrix
Post by TJMann on Apr 12th, 2008, 7:52am
I get zero with both matrices. What makes this possible is the integers mod 3 and mod 6 wrap around. In mod 3,  2+1 = 3 = 0, so 2+1 + 1+2 = 6 = 0.  In mod 6, 3+1+2 = 6, so (3+1+2) + (1+2+3) + (2+3+1) = 18 = 0.
Title: Re: matrix
Post by pex on Apr 12th, 2008, 8:14am
It's a bit of a stretch, but the following 2x1 matrix seems to fit the criteria:

[ 1-i ]
[ i-1 ]

- All numbers add up to zero: obvious.
- No negatives: nonreal complex numbers are neither "positive" nor "negative".
- No imaginary numbers: the numbers in this matrix are complex, but not imaginary (http://mathworld.wolfram.com/ImaginaryNumber.html), because the real parts are nonzero.
- No zeros: obvious.
- No "infinity or something like that": obvious.
Title: Re: matrix
Post by Michael Dagg on Apr 15th, 2008, 10:42pm
> But is 2 > 0 mod 3?

If you are asking if there is order in the
ring  Z_3  then the answer is no.  This in no
way invalidates addition modulo 3, including
multiplication; in fact, it is the other way around.
Title: Re: matrix
Post by Obob on Apr 16th, 2008, 8:10am
Yes, I slightly misread the original post.  By reading "no negatives, no imaginary numbers, and no 0s" I automatically concluded that meant "positive numbers."  You could either interpret imaginary numbers as "purely imaginary" numbers, as pex did, or do some modular arithmetic example and just say that the elements are not negative or imaginary, since those terms don't even make sense.

At any rate, though, this riddle is mostly a wad of nonsense.  At least the intended solution is.
Title: Re: matrix
Post by temporary on Apr 16th, 2008, 8:17pm

on 04/12/08 at 08:14:25, pex wrote:
It's a bit of a stretch, but the following 2x1 matrix seems to fit the criteria:

[ 1-i ]
[ i-1 ]

- All numbers add up to zero: obvious.
- No negatives: nonreal complex numbers are neither "positive" nor "negative".
- No imaginary numbers: the numbers in this matrix are complex, but not imaginary (http://mathworld.wolfram.com/ImaginaryNumber.html), because the real parts are nonzero.
- No zeros: obvious.
- No "infinity or something like that": obvious.

I might have misphrased it, but you did kind of use negative numbers anyway. I'll fix the misphrase to avoid confusing anyone.
Title: Re: matrix
Post by pex on Apr 17th, 2008, 1:27am
Okay... so we either need a nonstandard definition of addition, or a different set of numbers than real or complex numbers.

For example, in the 10-adic numbers, ...9999999 + 1 = 0. (The representation ...9999999 is infinite, but the number it represents isn't - does that count?)
Title: Re: matrix
Post by pex on Apr 17th, 2008, 4:34am
Just a thought that crossed my mind: is there a sensible way to define matrices of dimensions m x n, with m or n equal to zero? If yes, one might argue that the sum of its elements (an empty set) equals zero, and matrices of dimensions 0x0, 0x1, 1x0, 0x2, 2x0, 0x3, 3x0, ... would work.
Title: Re: matrix
Post by Michael Dagg on Apr 17th, 2008, 3:24pm
Of course I would have hardly guessed that
those kind of matrices would be involved in
solution a here, but some matrices like these can
pop up when finite groups are wrote using Smith
Normal Form, in particular, subgroups of
G = (Z/p^2 Z) x (Z/pZ)  for example.
Title: Re: matrix
Post by TJMann on Apr 17th, 2008, 5:37pm
Being curious but knowing a just little about groups, how do you make the product (Z/p^2Z) X (Z/pZ)?  I looked my algebra book about it but I did not see an explicit way to actually do it or should I think that it is just a "product" through the use of notation?
Title: Re: matrix
Post by Obob on Apr 17th, 2008, 7:58pm
Given two groups G and H, you form the product G X H as the set of all pairs (g,h), where g is in G and h is in H.  You multiply pairs (g1,h1) and (g2,h2) by the rule (g1,h1) * (g2,h2) = (g1 * g2,h1 * h2).  So this is totally explicit, but it is also essentially just taking the product through the use of notation.
Title: Re: matrix
Post by TJMann on Apr 18th, 2008, 5:05pm
Thx. I did know that and it seems to fit the idea that I thinking because if we have p = 2 then we have the product (Z/4Z) X (Z/2Z) which is more specific that the general form wrote but aren't both Z/4Z and Z/2Z finite fields? One is of length 4 and the other is of length 2 and now that means the product would effectively be a 4 x 2 - tuple?  Groups that I know about don't go that far as to require what I suppose are called lists.
Title: Re: matrix
Post by Obob on Apr 19th, 2008, 7:46pm
Z/4Z is not a field.  2 has no multiplicative inverse in Z/4Z.  But you can view Z/4Z x Z/2Z as a two-tuple of numbers, one coming from the set {0,1,2,3} and the other coming from {0,1}.  The group Z/4Z x Z/2Z does in fact have 8 elements.
Title: Re: matrix
Post by temporary on Apr 22nd, 2008, 5:11pm

on 04/17/08 at 04:34:02, pex wrote:
Just a thought that crossed my mind: is there a sensible way to define matrices of dimensions m x n, with m or n equal to zero? If yes, one might argue that the sum of its elements (an empty set) equals zero, and matrices of dimensions 0x0, 0x1, 1x0, 0x2, 2x0, 0x3, 3x0, ... would work.

Yep. That pretty much hits the nail on the head.
Title: Re: matrix
Post by Michael Dagg on Apr 22nd, 2008, 7:02pm
> Yep. That pretty much hits the nail on the head.

That's stated lightly -- got a proof?
Title: Re: matrix
Post by temporary on Apr 24th, 2008, 6:53pm
Isn't it obvious and self-proven. A matrix with no numbers has to add up to 0. And I am prepared to bet 1ud8f7g6g6v7v7v7 dollars that it does. And the only time I bet something I don't have is when it doesn't matter because I win.
Title: Re: matrix
Post by Michael Dagg on Apr 24th, 2008, 7:11pm
That is not a solution to the problem you posted.

It is technically clear, do you know why?
Title: Re: matrix
Post by temporary on Apr 24th, 2008, 7:26pm
There is an answer. A matrix with no numbers(like 0x3, 1x0, 9x0, 0x(pi+e), etc) Where have you been for the last several posts.
Title: Re: matrix
Post by Michael Dagg on Apr 24th, 2008, 7:42pm
A matrix with no numbers is not a solution to the
problem you posted. It is true that a sum on an
empty set is zero but it does not satisfy the
hypothesis of your problem.


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