Geodesic
On the Distance to the Closest Matrix with Triple Zero Eigenvalue
Matematičeskie zametki, Tome 73 (2003) no. 4, pp. 545-555.

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The 2-norm distance from a matrix $A$ to the set $\mathscr M$ of $(n\times n)$ matrices with a zero eigenvalue of multiplicity $\ge3$ is estimated. If $$ Q(\gamma_1,\gamma_2,\gamma_3)=\begin{pmatrix} A\gamma_1I_n\gamma_3I_n \\0\gamma_2I_n \\00 \end{pmatrix}, \qquad n\ge3, $$ then $$ \rho_2(A,\mathscr M) \ge\max_{\gamma_1,\gamma_2\ge0,\,\gamma_3\in\mathbb C} \sigma_{3n-2}(Q(\gamma_1,\gamma_2,\gamma_3)), $$ where $\sigma_i(\cdot)$ is the $i$th singular value of the corresponding matrix in the decreasing order of singular values. Moreover, if the maximum on the right-hand side is attained at the point $\gamma^*=(\gamma^*_1,\gamma^*_2,\gamma^*_3)$, where $\gamma^*_1\gamma^*_2\ne0$, then, in fact, one has the exact equality $$ \rho_2(A,\mathscr M) =\sigma_{3n-2}(Q(\gamma^*_1,\gamma^*_2,\gamma^*_3)). $$ This result can be regarded as an extension of Malyshev's formula, which gives the 2-norm distance from $A$ to the set of matrices with a multiple zero eigenvalue. 
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Kh. D. Ikramov; A. M. Nazari. On the Distance to the Closest Matrix with Triple Zero Eigenvalue. Matematičeskie zametki, Tome 73 (2003) no. 4, pp. 545-555. http://geodesic.mathdoc.fr/item/MZM_2003_73_4_a5/

[1] Malyshev A. N., “A formula for the $2$-norm distance from a matrix to the set of matrices with multiple eigenvalues”, Numer. Math., 83 (1999), 443–454 | DOI | MR | Zbl

[2] Demmel Dzh., Vychislitelnaya lineinaya algebra. Teoriya i prilozheniya, Mir, M., 2001

[3] Sun J.-G., “A note on simple non-zero singular values”, J. Comput. Math., 6 (1988), 259–266