The Segre image is the projective rank-one locus cut out by 2 by 2 minors

If $z_{ij}=x_i{y}_j$, then $z_{ij}{z}_{k\mathrm{\ell }}=x_i{x}_k{y}_j{y}_{\mathrm{\ell }}=z_{i\mathrm{\ell }}{z}_{kj}$, so every Segre point satisfies every displayed minor.

Conversely choose a nonzero entry $z_{i_0{j}_0}$. The minor equations give $z_{ij}{z}_{i_0{j}_0}=z_{i{j}_0}{z}_{i_{0}j}$, hence $z_{ij}=x_i{y}_j$ for $x_i=z_{i{j}_0}$ and $y_j=z_{i_{0}j}\mathrm{/}{z}_{i_0{j}_0}$. Thus the matrix is a Segre image.

The minors are homogeneous quadrics, so their common zero locus is projectively closed, and step 2.1 identifies it with the image. On $z_{i_0{j}_0}\mathrm{\ne }0$, the ratios $\frac{z_{i{j}_0}}{z_{i_0{j}_0}},\frac{z_{i_{0}j}}{z_{i_0{j}_0}}$ are regular and recover the standard affine coordinates of both factors. Thus the asserted chartwise inverse is proved directly; injectivity alone is not used as an embedding criterion.