Consider the singular value decompositions $A=U_A\Sigma_AV_A^T$ and $B=U_B\Sigma_BV_B^T$. Is there a word that describes the relation between $A$ and $B$ when they have the same left and/or right singular vectors ($U_A=U_B$ and/or $V_A=V_B$)?
If we had $\Sigma_A=\Sigma_B$ instead, then we would say that the matrices are unitarily equivalent . If we had the eigenvalue decomposition instead of the SVD, we would say the matrices are simultaneously diagonalisable. I am looking for a similar concise and accepted term applicable to the situation I described.
Somewhat related: Does Sharing same singular vectors lead to some properties?