Re: Matrix equation XA+AX=B

On 10.04.2008 20:09, Ian Parker wrote:
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.


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

Counterexample for n=2: A=((0 1),(0 0)) (read row-wise) and T = transpose(A).

Best wishes,
J.

[ Moderator's note: I'm not sure what Ian Parker meant by "we can see from inspection that the T is symmetric", but perhaps it's that if A
is symmetric the coefficient matrix for the system of equations is symmetric. Moreover, if both A and T are symmetric, then transpose(X) is a solution if and only if X is, and in that case we need only consider symmetric solutions X.

Any condition for the existence and uniqueness of solutions should only involve the homogeneous problem, and thus the matrix A, not T. In particular,
a necessary (but not sufficient) condition is that A is nonsingular.

-RI ]
.

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