Re: [math-fun] Name for matrices whose rows & columns both sum to zero ?
About the only analogies I can think of are parity check codes, where the row/column sums are mod 2, or higher-order codes where the sums are in GF(2^k). But my original posting was about standard arithmetic over Q, R, C, etc. At 07:39 PM 4/28/2013, Victor Miller wrote:
I was thinking that too, but the elements of a doubly stochastic matrix are non-negative (since they correspond to probabilities).
Victor
On Sun, Apr 28, 2013 at 10:33 PM, Michael Kleber <michael.kleber@gmail.com>wrote:
In the square case, you can add 1/n to each entry and make each row and column sums to 1, a "doubly-stochastic matrix", about which much has been written. (The dimension of the space is certainly (n-1)^2, for example.) But I don't know of a name for the sum-to-0 version.
--Michael
On Sun, Apr 28, 2013 at 10:04 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
Though it's not (quite) obvious that element [n, n] is not over-determined ... WFL
On 4/29/13, Henry Baker <hbaker1@pipeline.com> wrote:
Woops! At least one of these equations is redundant, so perhaps the dimension is n^2-2n-1 = (n-1)^2.
(This part is obvious: take a random (n-1)x(n-1) matrix & pad it with one additional row & column, which consists of -row sums and -column sums.)
At 05:21 PM 4/28/2013, Henry Baker wrote:
Is there a name for matrices whose rows & columns both sum to zero ?
What about _square_ matrices of this type ?
Clearly, there are n^2 unknowns and 2n equations, so the nxn matrices have dimension n^2-2n = n(n-2).
The difference between two doubly stochastic matrices would have zero sums for rows and columns. Brent Meeker On 4/28/2013 9:41 PM, Henry Baker wrote:
About the only analogies I can think of are parity check codes, where the row/column sums are mod 2, or higher-order codes where the sums are in GF(2^k). But my original posting was about standard arithmetic over Q, R, C, etc.
At 07:39 PM 4/28/2013, Victor Miller wrote:
I was thinking that too, but the elements of a doubly stochastic matrix are non-negative (since they correspond to probabilities).
Victor
On Sun, Apr 28, 2013 at 10:33 PM, Michael Kleber <michael.kleber@gmail.com>wrote:
In the square case, you can add 1/n to each entry and make each row and column sums to 1, a "doubly-stochastic matrix", about which much has been written. (The dimension of the space is certainly (n-1)^2, for example.) But I don't know of a name for the sum-to-0 version.
--Michael
On Sun, Apr 28, 2013 at 10:04 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
Though it's not (quite) obvious that element [n, n] is not over-determined ... WFL
On 4/29/13, Henry Baker <hbaker1@pipeline.com> wrote:
Woops! At least one of these equations is redundant, so perhaps the dimension is n^2-2n-1 = (n-1)^2.
(This part is obvious: take a random (n-1)x(n-1) matrix & pad it with one additional row & column, which consists of -row sums and -column sums.)
At 05:21 PM 4/28/2013, Henry Baker wrote:
Is there a name for matrices whose rows & columns both sum to zero ?
What about _square_ matrices of this type ?
Clearly, there are n^2 unknowns and 2n equations, so the nxn matrices have dimension n^2-2n = n(n-2).
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For the square case over a field K: Let J = [1,1,...,1]. Then you are looking at the set of nxn matrices A such that J.A = 0 and A.Transpose(J) = 0. This set is closed under linear combinations and multiplication. So it is a finite dimensional linear associative algebra. With some effort it may be possible to classify these algebras using the Wedderburn-Artin Theorems. Of course if you do this you lose the particular representation given. I can only say that if the field is K and n = 2 the algebra is isomorphic to K since it is 1-dimensional and has identity [1/2,-1/2; -1/2,1/2]. What's the 4 dimensional algebra you get when n = 3? --Edwin On Mon, Apr 29, 2013 at 12:41 AM, Henry Baker <hbaker1@pipeline.com> wrote:
About the only analogies I can think of are parity check codes, where the row/column sums are mod 2, or higher-order codes where the sums are in GF(2^k). But my original posting was about standard arithmetic over Q, R, C, etc.
At 07:39 PM 4/28/2013, Victor Miller wrote:
I was thinking that too, but the elements of a doubly stochastic matrix are non-negative (since they correspond to probabilities).
Victor
On Sun, Apr 28, 2013 at 10:33 PM, Michael Kleber <michael.kleber@gmail.com>wrote:
In the square case, you can add 1/n to each entry and make each row and column sums to 1, a "doubly-stochastic matrix", about which much has been written. (The dimension of the space is certainly (n-1)^2, for example.) But I don't know of a name for the sum-to-0 version.
--Michael
On Sun, Apr 28, 2013 at 10:04 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
Though it's not (quite) obvious that element [n, n] is not over-determined ... WFL
On 4/29/13, Henry Baker <hbaker1@pipeline.com> wrote:
Woops! At least one of these equations is redundant, so perhaps the dimension is n^2-2n-1 = (n-1)^2.
(This part is obvious: take a random (n-1)x(n-1) matrix & pad it with one additional row & column, which consists of -row sums and -column sums.)
At 05:21 PM 4/28/2013, Henry Baker wrote:
Is there a name for matrices whose rows & columns both sum to zero ?
What about _square_ matrices of this type ?
Clearly, there are n^2 unknowns and 2n equations, so the nxn matrices have dimension n^2-2n = n(n-2).
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