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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 $N$?

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  • $\begingroup$ Where does this problem come from? Turning a $d\times d$ matrix into a $1 \times d^2$ row doesn't seem very natural. $\endgroup$
    – Arnaud Mortier
    Aug 4, 2019 at 20:59
  • $\begingroup$ The resulting matrix is $n \times d^2$.. but each row is $d^2$ dimensional. $\endgroup$
    – asdf
    Aug 4, 2019 at 22:33
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    $\begingroup$ 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 $d^2 \leq n$? $\endgroup$
    – Ben Grossmann
    Aug 4, 2019 at 23:17

1 Answer 1

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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 $$ N = \pmatrix{ \operatorname{vec}(m_1m_1^T)^T \\ \vdots\\ \operatorname{vec}(m_nm_n^T)^T} = \pmatrix{[m_1 \otimes m_1]^T\\ \vdots \\ [m_n \otimes m_n]^T} = \pmatrix{[e_1 \otimes e_1]^T\\ \vdots\\ [e_n \otimes e_n]^T}(M \otimes M) $$

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