Thread Subject: Solving Equation

Hi,

How do I solve for X in equation

X - MXM' = Q

where M, X, and Q are all (n,n) matrix.

Thanks.

sumnaray@gmail.com wrote in message
<d2ea35f2-b295-4b24-a58f-4aa7bf3359ae@s8g2000prg.googlegroups.com>...
> Hi,
>
> How do I solve for X in equation
>
> X - MXM' = Q
>
> where M, X, and Q are all (n,n) matrix.
>
> Thanks.

See thread on similar problem here (Q was identity)

http://www.mathworks.com/matlabcentral/newsreader/view_thread/159258

Bruno

On Nov 23, 2:58 am, "Bruno Luong" <brunolu...@yahoo.com> wrote:
> sumna...@gmail.com wrote in message
>
> <d2ea35f2-b295-4b24-a58f-4aa7bf335...@s8g2000prg.googlegroups.com>...
>
> > Hi,
>
> > How do I solve for X in equation
>
> > X - MXM' = Q
>
> > where M, X, and Q are all (n,n) matrix.
>
> > Thanks.
>
> See thread on similar problem here (Q was identity)
>
> http://www.mathworks.com/matlabcentral/newsreader/view_thread/159258
>
> Bruno

Actually, I did try that, but I got
   NaN NaN NaN 0
   NaN NaN NaN 0
   NaN NaN -Inf 0
   NaN NaN 0 1

I did not know what to do after that. :-S

"Bruno Luong" <brunoluong@yahoo.com> wrote in message
<fi6181$8dm$1@fred.mathworks.com>...
> sumnaray@gmail.com wrote in message
>
<d2ea35f2-b295-4b24-a58f-4aa7bf3359ae@s8g2000prg.googlegroups.com>...
> > Hi,
> >
> > How do I solve for X in equation
> >
> > X - MXM' = Q
> >
> > where M, X, and Q are all (n,n) matrix.
> >
> > Thanks.
>
> See thread on similar problem here (Q was identity)
>
>
http://www.mathworks.com/matlabcentral/newsreader/view_thread/159258
>

Adapted program:

Bruno
--------------------------

n=5;

M=rand(n);
q=rand(n);
Q=q'*q

Mnlist=cell(1,n);
Mnlist(:)={M'};

BIGM1=blkdiag(Mnlist{:})';
BIGM2=reshape(permute(reshape(BIGM1,n,n,n,n),[2 1 4
3]),n*n,n*n);
X=reshape((eye(n*n)-BIGM1*BIGM2)\Q(:),n,n)
X - M*X*M'

On Nov 23, 3:17 am, "Bruno Luong" <brunolu...@yahoo.com> wrote:
> "Bruno Luong" <brunolu...@yahoo.com> wrote in message
>
> <fi6181$8d...@fred.mathworks.com>...> sumna...@gmail.com wrote in message
>
> <d2ea35f2-b295-4b24-a58f-4aa7bf335...@s8g2000prg.googlegroups.com>...> > Hi,
>
> > > How do I solve for X in equation
>
> > > X - MXM' = Q
>
> > > where M, X, and Q are all (n,n) matrix.
>
> > > Thanks.
>
> > See thread on similar problem here (Q was identity)
>
> http://www.mathworks.com/matlabcentral/newsreader/view_thread/159258
>
>
>
> Adapted program:
>
> Bruno
> --------------------------
>
> n=5;
>
> M=rand(n);
> q=rand(n);
> Q=q'*q
>
> Mnlist=cell(1,n);
> Mnlist(:)={M'};
>
> BIGM1=blkdiag(Mnlist{:})';
> BIGM2=reshape(permute(reshape(BIGM1,n,n,n,n),[2 1 4
> 3]),n*n,n*n);
> X=reshape((eye(n*n)-BIGM1*BIGM2)\Q(:),n,n)
> X - M*X*M'

Thanks for your code.

But I am afraid, I still get NaN in my result. I am trying this with

M = [0.5 -0.5 0.5 0; 0 1 0 0; -0.5 -0.5 1.5 0; 0 0 0 0]
and Q as eye(4);

And if you mind me asking, what does [2 1 4 3] mean in BIGM2
expression? I am sorry, I am not very familiar with Matlab.

Thanks again.

Why I bother to symmetrize input Q? Here is the program for
general Q:

n=5; % dimension

% Input
M=rand(n);
Q=rand(n)

% Solve
Mnlist=cell(1,n);
Mnlist(:)={M};
BIGM1=blkdiag(Mnlist{:});
BIGM2=reshape(permute(reshape(BIGM1,n,n,n,n),[2 1 4
3]),n*n,n*n);
X=reshape((eye(n*n)-BIGM1*BIGM2)\Q(:),n,n)

% Check
X - M*X*M'

sumnaray@gmail.com wrote in message

>
> But I am afraid, I still get NaN in my result. I am trying
this with
>
> M = [0.5 -0.5 0.5 0; 0 1 0 0; -0.5 -0.5 1.5 0; 0 0 0 0]

In order the problem to be well-posed, the input M needs to
be a full-rank matrix (inversible). Your matrix isn't. That
is why you get NaN.


>
> And if you mind me asking, what does [2 1 4 3] mean in BIGM2
> expression? I am sorry, I am not very familiar with Matlab.
>

I get an intermediate 4-D array which I want to reshape by
swapping dimensions 1 with 2; and also 3 with 4.

Bruno

sumnaray@gmail.com wrote in message <d2ea35f2-b295-4b24-
a58f-4aa7bf3359ae@s8g2000prg.googlegroups.com>...
> Hi,
>
> How do I solve for X in equation
>
> X - MXM' = Q
>
> where M, X, and Q are all (n,n) matrix.
>
> Thanks.
-------
  This should do it:

U = repmat(M,n,n);
K = floor((n:n^2+n-1)/n);
V = conj(M(K,K));
X = reshape((eye(n^2)-U.*V)\Q(:),n,n);

If M is real-valued, the 'conj' operation is unnecessary.

Roger Stafford

"Bruno Luong" <brunoluong@yahoo.com> wrote in message
<fi63hb$2o0$1@fred.mathworks.com>...
> sumnaray@gmail.com wrote in message
>
> >
> > But I am afraid, I still get NaN in my result. I am trying
> this with
> >
> > M = [0.5 -0.5 0.5 0; 0 1 0 0; -0.5 -0.5 1.5 0; 0 0 0 0]
>
> In order the problem to be well-posed, the input M needs to
> be a full-rank matrix (inversible). Your matrix isn't. That
> is why you get NaN.
>

Wait a minute, I'm not certain about this statement. I think
that the condition on M for well-posed is more complex than
that.

Bruno