Geodesic
On the 2-norm distance from a normal matrix to the set of matrices with a multiple zero eigenvalue
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XVIII, Tome 323 (2005), pp. 50-56 Cet article a éte moissonné depuis la source Math-Net.Ru

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An elementary proof is given for a formula for the 2-norm distance from a normal matrix $A$ to the set of matrices with a multiple zero eigenvalue. Earlier, the authors obtained this formula as an implication of a nontrivial result due to A. N. Malyshev. 
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Kh. D. Ikramov; A. M. Nazari. On the 2-norm distance from a normal matrix to the set of matrices with a multiple zero eigenvalue. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XVIII, Tome 323 (2005), pp. 50-56. http://geodesic.mathdoc.fr/item/ZNSL_2005_323_a5/

[1] A. N. Malyshev, “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] Kh. D. Ikramov, A. M. Nazari, “Ob odnom zamechatelnom sledstvii formuly Malysheva”, DAN, 385 (2002), 599–600 | MR

[3] Kh. D. Ikramov, “Yavnye formuly dlya matritsy s kratnym sobstvennym znacheniem nul, blizhaishei k zadannoi normalnoi matritse”, DAN, 398 (2004), 599–601 | MR

[4] C. Eckart, G. Young, “The approximation of one matrix by another of lower rank”, Psychometrica, 1 (1936), 211–218 | DOI | Zbl