[math-fun] Name for matrices whose rows & columns both sum to zero ?
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).
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).
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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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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-- Forewarned is worth an octopus in the bush.
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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-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (4)
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Fred lunnon -
Henry Baker -
Michael Kleber -
Victor Miller