I saw this statement in a lecture note



> 
> Assume the generalized SVD of matrices $A\in R^{m\times n}$ and $B\in R^{p\times n}$ given as:
> 
> 
> $$U^TAX = diag(\alpha\_1, ..., \alpha\_n),~ U^TU = I\_m$$
> $$V^TBX = diag(\beta\_1, ..., \beta\_q),~ V^TV = I\_p, ~q = min\{p, n\}$$
> 
> 
> 


I found that the right singular value vectors of the SVD of matrices $A$ and $B$ here are a same matrix $X$. I don't know how to find the SVD like that. Is that any two matrices with the same number of columns can be decomposed via this way?



SVD of two matrices A and B having the same right singular vectors?
Those notes do not refer to the "usual" SVD, but to the [**generalized SVD**](https://en.wikipedia.org/wiki/Generalized_singular_value_decomposition), which is a different decomposition of a pair of matrices (and does not require $X$ to be orthogonal, in particular).


For a quick introduction, you can check the Golub-Van Loan book, for instance.


