Re: Matrix equation XA+AX=B

On 9 Apr, 21:47, sanyi <vessz...@xxxxxxxxx> wrote:
In fact I have the equation  XA+AX=ATA. What I need is some necessary
and/or sufficient conditions for the existence, uniqueness of the
solution X.
Many thanks for help,   Sandor

I think you have is this

XA (i,j) will be Sigma(j) X(i,j)A(j,i) + Sigma(j) A(i,j)X(j,i)

AT will be Sigma(j) A(i,j)T(j,i)
ATA Sigma(i,j)A(i,j)A(j,i)T(j,i)= Sigma(j) X(i,j)A(j,i) +
Sigma(j) A(i,j)X(j,i)

This gives us a set of equations to be solved. At forst sight we can
see that there are

n^2 equations and n^2 unknowns so unless we can detect singularity the
solution is existant and unique.

However we can see from inspection that the T is symmetric giving n(n
+1)/2 unknowns.

Another pointer. We can tell if a matrix is singular since we can
arrive at identical rows / colums by addition and multiplication. I
don't think the n(n+1)/2 dimensional matrix is although there may be
symmetry considerations that dictate it.

What I suggest you do is use a random number generator to construct
some 10*10 matrices. Then construct a 55*55 matrix and try Gaussian
elimination. These are small by computational standards. If the matrix
is non singular its solution will be existant and unique.

In fact if a solution exists it will be unique.


- Ian Parker

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