# Relation between condition number of two related matrices

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 $$N$$?

• Where does this problem come from? Turning a $d\times d$ matrix into a $1 \times d^2$ row doesn't seem very natural. Aug 4, 2019 at 20:59
• The resulting matrix is $n \times d^2$.. but each row is $d^2$ dimensional.
– asdf
Aug 4, 2019 at 22:33
• If $d^2 > n$, then $N$ is a wide matrix which means that (by the usual definition) $N$ will necessarily have a zero singular value, which means that $\kappa$ will be undefined (or infinite, if you prefer). Should we consider only the case where $d^2 \leq n$? Aug 4, 2019 at 23:17

In terms of the vectorization operator and Kronecker products, we could say that $$N = \pmatrix{ \operatorname{vec}(m_1m_1^T)^T \\ \vdots\\ \operatorname{vec}(m_nm_n^T)^T} = \pmatrix{[m_1 \otimes m_1]^T\\ \vdots \\ [m_n \otimes m_n]^T} = \pmatrix{[e_1 \otimes e_1]^T\\ \vdots\\ [e_n \otimes e_n]^T}(M \otimes M)$$