Relation between sum of matrices and norm.

Question

Let $M \in \mathcal{M}_n(\mathbb{R})$ invertible. Find a matrix $N$ such that $M + N$ is not invertible and $\|N\|_2$ is minimum.

Is easy to find matrices $N$ such that $M + N$ is not invertible (in fact, there is some results that give needed and sufficient conditions for that). My problem is with the requirement "$\|N\|_2$ is minimum". I really have no ideia about how to approach this problem. I appreciate any hints.

Answer 1

You can use the SVD decomposition of $M$ $$M = U \Sigma V^T = \sum_{i=1}^{n}{\lambda_i u_i v_i^T}$$

Where the $\lambda_i$ are the singular values (real positive), the $u_i$ are the left-singular vectors and the $v_i$ are the right singular vectors.

Assuming $\lambda_n$ is the minimum singular value, then the minimum $N$ matrix such that $M + N$ is singular is:

$$N = - \lambda_n u_n v_n^T$$

Answer 2

Hint. Suppose $(M+N)x=0$ for some unit vector $x$. Then $\|N\|_2\ge\|Nx\|_2=\|Mx\|_2\ge\sigma_n(M)$.