I have the following problem: I have a matrix $M\in \mathbb{R}^{3\times 3}$ and I consider two SVD's $U_1DV_1^T$ and $U_2DV_2^T$ of $M$ with $D = \mathrm{diag}(\lambda_1,\lambda_1,\lambda_2)$. Therefore, the first and second column vector of $U_1$ are rotated with a rotation matrix $R$ $$U_2=[Ru_1^1,Ru_1^2,u_1^3]$$ because we have a two dimensional eigenspace. What is the relationship between $V_1$ and $V_2$? Is there also (the same) rotation matrix which rotates the first two rows or columns? Many greetings and thanks in advance.
3
-
$\begingroup$ I thought the SVD is unique (up to phase factors) if the $\lambda$'s are distinct? $\endgroup$– Carlo BeenakkerCommented Feb 11, 2020 at 12:30
-
$\begingroup$ The first two eigenvalues are equal, so the $\lambda$'s are not distinct. $\endgroup$– Palpatine2357Commented Feb 11, 2020 at 13:53
-
$\begingroup$ shouldn't you be rotating the rows of $U_1$, rather the columns? since you are inserting the rotation matrix beween $U_1$ and $D$, so it acts on the second index of $U_1$; and then the transpose of $R$ acts on the columns of $V_1^T$, and therefore on the rows of $V_1$. $\endgroup$– Carlo BeenakkerCommented Feb 11, 2020 at 15:10
Add a comment
|