Let $M \in \mathbb{R}^{n \times d}$, $m_i$ is a $i$-th row of $M$, and $\kappa(M)$ be the ratio between the biggest and the smallest singular values.
We define $N \in \mathbb{R}^{n \times d^2}$, where each row of $N$ is defined as $m_im_i^T$ (i.e. the outer product of a row).
What can we say on the condition number of $N$?
Not a complete answer, but too long to (conveniently) make into a comment.
In terms of the vectorization operator and Kronecker products, we could say that $$ N = \pmatrix{ \operatorname{vec}(m_1m_1^T)^T \\ \vdots\\ \operatorname{vec}(m_nm_n^T)^T} = \pmatrix{[m_1 \otimes m_1]^T\\ \vdots \\ [m_n \otimes m_n]^T} = \pmatrix{[e_1 \otimes e_1]^T\\ \vdots\\ [e_n \otimes e_n]^T}(M \otimes M) $$