Singular vectors of $AB^{\top} $ versus $B^{\top} A$

Question

Let $A$ and $B$ be matrices of dimensions $d \times n$. Let $C = AB^{\top}$.

We also know that $C = I \mathrm{diag}(\gamma) J$ for some matrices $I$ and $J$ and vector $\gamma$ of length $m$, $m < \min(d,n)$ (i.e. $I$ is of dimension $d \times m$ and $J$ is of dimension $m \times d$).

Using only the matrix $B^{\top}A$ (and not $AB^{\top}$, $I$, $J$ or $\gamma$), I want to find $U$ and $V$ of dimensions $m \times d$ such that $U I$ and $V J$ are invertible and $U A$ and $V B$ can be calculated. You can apply any decomposition or extract any information you need from $B^{\top} A$.

FIX: I also have $B^{\top} B$ and $A^{\top} A$ at my disposal.

EDIT: I managed to refine the question.

Let $\sigma(D)$ be the non-zero eigenvalues of a square matrix $D$ and let $s(C)$ be the non-zero singular values of a matrix $C$.

We know that if $C = AB^{\top}$, then:

$s^2(C) = \sigma(C C^{\top}) = \sigma(AB^{\top} B A^{\top}) = \sigma(A^{\top} A B^{\top} B)$

I am also assuming that I can compute $A^{\top} A$ and $B^{\top} B$, which means that $s^2(C)$ is computable.

I think $U$ and $V$ that I am looking for could come from the right and left singular vectors of $C$, but I don't know how to compute these singular vectors based on $A^{\top} A$, $B^{\top} B$ and $B^{\top} A$. Worst even, even if I had these $U$ and $V$, it is not clear how to compute $UA$ and $VB$ without explicitly representing $A$ and $B$.

EDIT 2: I think I managed to refine the question some more:

Assume we have the matrix equation:

$(U^{\top} A) (Q \Lambda Q^{\top}) (A^{\top} U) = \Sigma$

such that all matrices in this equation have real values and ($d > n$):

$U$ is $d \times d$ and is orthonormal

$A$ is $d \times n$

$Q$ is $n \times n$ and is orthonormal

$\Lambda$ is $n \times n$ and is diagonal (with only non-negative values)

$\Sigma$ is $d \times d$ and is diagonal (with only non-negative values)

Is there a way to extract the matrix $U^{\top} A$ if we know: $\Sigma$, $\Lambda$ and $Q$ but don't know $A$, $U$? (I think this can be phrased more "mathematically" as a question about a uniqueness of a solution, but not sure how.)

I wouldn't mind getting $U^{\top} A$ up to an invertible linear transformation, as long as this transformation is a function of $U$ only. What if $A$ contains $n$ orthonormal vectors of length $d$? I think this has an easier solution.

Answer 1

I don't think that there is an easy relation. The reason behind my thinking is that even for the following "nice" case, we have only a weak relation.

Let $A$ and $B$ be matrices for which the product $AB^T$ is normal (i.e., $BA^TAB^T = AB^TBA^T$). Then,

$$\sigma(AB^T)\quad \prec_w\quad \sigma(B^TA),$$

where $\prec_w$ denotes the weak majorization relation, and $\sigma(\cdot)$ denotes the vector of singular values sorted in descending order.