A real matrix $\mathbf M$ with square size $N\times N$ and rank $r$ can be projected into its (range) image $\operatorname {im}\left( \mathbf M\right)$ and (nullspace) kernel $\operatorname {ker}\left( \mathbf M\right)$. Usually, the projection is a square matrix $\mathbf P$ with $\mathbf P^2 = \mathbf P $. However, suppose we want to reduce the dimensionality, so that we wish to write the projection of $\mathbf M$ in the image space, call it $\overline{\mathbf M}$, which has dimension $r\times r$.
Thus, we need a map from ${\mathbb{R}}^{N\times N} \mapsto{\mathbb{R}}^{r\times r} $. What is this map called? It isn't a projection, since $\mathbf P^2 \neq \mathbf P $.
It is easy to find from the eigendecomposition of $\mathbf M$ as all the column eigenvectors with non-zero eigenvalues, which are part of an orthonormal basis of ${\mathbb{R}}^{N\times N}$. This transform operates on the basis vector space of $\mathbf M$. Call the transform $\mathbf{T}$, it can be chosen as any scaled version of these vectors and has the size ${\mathbb{R}}^{N\times r}$. It also has some properties since these vectors are part of an orthonormal basis.
The reduced matrix is $\mathbf{\overline M} =\mathbf{T}^{\intercal}\mathbf{ MT} $.
Could someone elaborate on this linear map on the basis space and what it is called?