I am looking for a method to solve the matrix equation $$ DXa = Xb $$ where $D\in \mathbb{R}^{n\times n}$ is diagonal, $a, b\in \mathbb{R}^{n}$ and $X$ is the unknown orthogonal $n\times n$ matrix such that $X^TX = I$. The matrices $a$ and $b$ have unit norm, and $D$ has diagonal entries of unit magnitude, so that the matrix equation is consistent.
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Your belief that there should always be a solution is wrong for $n=1$ if $D=a=1$ and $b=-1$, which satisfies all your requirements. Thus it seems unlikely that one can say anything in general. Posing the linear and the orthogonality constraint as a least squares problems in the Frobenius norm, and submitting it to an unconstrained optimization routine is probably the only thing one can do. |
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Generalizing Arnold's example, suppose $D = I$. Then the problem is to find an orthogonal $X$ satisfying $X(a-b) = 0$, which is possible only when $a = b$. Interestingly, in 2D with $D = \text{diag}(1, -1)$, one can always find a plane rotation $X$ that does the job. Evidently the solvability for arbitrary unit-norm $a$, $b$ depends only on $D$. What is the general condition that makes the problem solvable? Is it something like $D$ has negative determinant, i.e., it has an odd number of -1's on the diagonal? |
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