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From: Andreea Onea on 5 Jun 2010 13:59
Hello all,
i would like to ask you if there is any way to get an analytical (not numerical) solution for a system in the following form: X'*X=A, where A (pxp) known Matrix and X(nxp) the matrix i want to compute. I would also like to ask you what can be the maximum dimensions of X (max(n) & max(p)) so as to get a solution without having an overload.
Thanks in advance
From: Bruno Luong on 5 Jun 2010 16:16
"Andreea Onea" <onea_mihaela(a)yahoo.com> wrote in message <hue399$6mi$1(a)fred.mathworks.com>...
> Hello all,
> i would like to ask you if there is any way to get an analytical (not numerical) solution for a system in the following form: X'*X=A, where A (pxp) known Matrix and X(nxp) the matrix i want to compute.

The solution - if exists - is not unique, you can multiply left and/or right side of X by any hermitian matrix and it's still be a solution. Of course, there are an infinity of hermitian matrices, and they form a very interesting mathematics object studied intensively by human (e.g., Lie group), and have ax extremely important role in theoretical physics at our time.

If you want a particular solution numerically, look at SVD or SQRTM.

As the solution is not unique your question of "can I express X as ..." does not make sense as long as you do not specify which solution you want. I guess there is always away to express SQRTM by formula of the elements of A, but the formula would perhaps not be fit into 1000 pages, and needless to say useless.

>I would also like to ask you what can be the maximum dimensions of X (max(n) & max(p)) so as to get a solution without having an overload.

You can select X with the number of row n as large as you like. I don't see any reason why it should be any how limited.

Bruno
From: Andreea Onea on 5 Jun 2010 18:07
"Bruno Luong" <b.luong(a)fogale.findmycountry> wrote in message <hueba4$hnt$1(a)fred.mathworks.com>...
> "Andreea Onea" <onea_mihaela(a)yahoo.com> wrote in message <hue399$6mi$1(a)fred.mathworks.com>...
> > Hello all,
> > i would like to ask you if there is any way to get an analytical (not numerical) solution for a system in the following form: X'*X=A, where A (pxp) known Matrix and X(nxp) the matrix i want to compute.
>
> The solution - if exists - is not unique, you can multiply left and/or right side of X by any hermitian matrix and it's still be a solution. Of course, there are an infinity of hermitian matrices, and they form a very interesting mathematics object studied intensively by human (e.g., Lie group), and have ax extremely important role in theoretical physics at our time.
>
> If you want a particular solution numerically, look at SVD or SQRTM.
>
> As the solution is not unique your question of "can I express X as ..." does not make sense as long as you do not specify which solution you want. I guess there is always away to express SQRTM by formula of the elements of A, but the formula would perhaps not be fit into 1000 pages, and needless to say useless.
>
> >I would also like to ask you what can be the maximum dimensions of X (max(n) & max(p)) so as to get a solution without having an overload.
>
> You can select X with the number of row n as large as you like. I don't see any reason why it should be any how limited.
>
> Bruno


Thank you very much for your answer.To be honest, as I didn't give a complete information in my first post, your solution could not work in my case because both SVD AND SQRTM, as far as I know, try to find the least square solutions.

To be more specific I have the matrix A of order k x k. From A I want to go back to find (if it exists, of course) the matrix X, which is a matrix with elements only +1 or -1 (of order n x k, n and k being known from the beginning). I do this through the formula A = X' * X, where X' is the transpose of X.
When i try to construct the system by hand, I arrived to an non-linear system with n quadratic equations (see example below). So, I think it could be unrealistic to insert n equations in matlab and solve my system.
Moreover, I tried to construct a function F = X'X-A and try to solve it using FSOLVE command in Matlab but didn't give me any result, as this command is working only when X is square matrix and usually my X is not.

Example:

A = [7 -1 -1; -1 7 -1; -1 -1 7] (3x3)

I want to find X = 7x3 matrix:
X' = [a11 a12 a13 a14 a15 a16 a17
a21 a22 a23 a24 a25 a26 a27
a31 a32 a33 a34 a35 a36 a37]

then the system when A = X' * X would be:

sum(ai1^2) = sum(ai2^2) = sum (ai3^2) = 7 , i = 1,...,7
sum(ai1 * ai2) = sum(ai1 * ai3) = sum(ai2 * ai3) = -1, i = 1,...,7

a_ij = +1 or -1, i = 1,..7, j = 1,...,3

Thank you again. Any help would be greatly appreciated.
Andreea
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