Consider the case of a transfer matrix with 3 inputs and 2 outputs, and suppose I perform the singular value decompusition of it. If I do this, I will have that the matrix of the inputs $V$ will be a $3\times3$ matrix, the matrix of the singular values will be $2\times3$ and the matrix of the outputs will be a $2\times2$ matrix.
In particular, the matrix of the singular value will have the last column to be equal to all zeros, for example look at here: https://ocw.mit.edu/courses/aeronautics-and-astronautics/16-323-principles-of-optimal-control-spring-2008/lecture-notes/lec14.pdf at pag. 7, where there is a similar example, which is:
How do I interpret the fact that the last column of the matrix of singular value is all zeros?
I know that it is done for reasons of computations, but from a control systems point of view, how do I interpret this?
Does it means that there is an input which has zero output?
You can think about a matrix with column $i$ all zeros as a matrix that projects every vector onto the hyperplane where the $i^{th}$ dimension is zero.
Your $2 \times 3$ matrix is a linear transformation from $\mathbb{R}^3$ to $\mathbb{R}^2$, so you're sending all vectors to the $xy$-plane, i.e. where $z=0$.
Similarly, a $3 \times 2$ matrix with a zero third row is a linear transformation from $\mathbb{R}^2$ to $\mathbb{R}^3$ where the vectors remain in the $xy$-plane, but now live in 3D space.
Is that what you were looking for?