# What is the role of left singular vectors in SVD?

SVD decomposition of a data matrix $$A$$ of order $$n \times d$$ and rank $$r$$ can be expressed as follows

$$A_{n\times d} = U_{n\times r}D_{r \times r}V^{T}_{r \times d}$$

The rows of the data matrix $$A$$ are the data points in $$d$$ dimensional space. Thus, there are $$n$$ points in $$d$$ dimensional space.

The matrix $$V$$ contains $$r$$ right singular vectors as columns. Right singular vectors are orthonormal and forms a $$r-$$dimensional subspace that best fits the given $$n$$ data points.

The matrix $$D$$ is a diagonal matrix that contains singular values. Singular values signify the least squares (loss) of n-data points on the subspace $$r$$ right singular vectors.

The matrix $$U$$ contains left singular vectors as columns. Left singular vectors are also orthonormal.

But what does the $$r$$ left singular vectors signify?