Journal of Mathematical Sciences, Vol. 137, No. 3, 2006 ON THE 2-NORM DISTANCE FROM A NORMAL MATRIX TO THE SET OF MATRICES WITH A MULTIPLE ZERO EIGENVALUE Kh. D. Ikramov ∗ and A. M. Nazari ∗ UDC 512 An elementary proof of a formula for the 2-norm distance from a normal matrix A to the set of matrices with a multiple zero eigenvalueis given. Earlier, the authors obtained this formula as an implication of a nontrivial result due to A. N. Malyshev. Bibliography: 4 titles. 1. Let M n (C) be the set of n × n complex matrices (n ≥ 2) and let L be its subset consisting of matrices with a multiple zero eigenvalue. With a matrix A ∈ M n (C) we associate the matrix P (γ)=  A γI n 0 A  of order 2n, which depends on the real parameter γ. In [1], for the 2-norm distance ρ 2 (A, L) from A to the set L the following formula was obtained: ρ 2 (A, L) = max γ≥0 σ 2n−1 (P (γ)). (1) Here, σ 2n−1 (P (γ)) is the next-to-the-last singular value of the matrix P (γ), and the singular values are arranged nonincreasingly. Let A ∈ M n (C) be a normal matrix with the eigenvalues λ 1 ,...,λ n , numbered is such a way that |λ 1 |≥|λ 2 |≥ ... ≥|λ n |. In [2], it was shown that ρ 2 (A, L)=  |λ n−1 | 2 + |λ n | 2 2  1/2 . (2) The proof of this relation in [2] was based on the use of formula (1). Finally, explicit formulas for computing the matrix B = A +∆ ∈L closest (with respect to the 2-norm) to a normal matrix A were given in [3]. These formulas have the following remarkable feature: the correction matrix ∆ is determined by the two eigenvalues of A smallest in absolute value (i.e., by λ n−1 and λ n ) and the corresponding eigenvectors q n−1 and q n . This implies that the problem of approximating a normal matrix by a matrix belonging to L is essentially two-dimensional and, probably, an elementary proof of formula (2) exists. The proof given in [2] is based on the nontrivial formula (1), and it is by no means elementary. The purpose of this short paper is to propose an elementary proof, which uses only simple facts from calculus. 2. From the above discussion it follows that we can confine ourselves to considering normal matrices of second order. Since the 2-norm is unitarily invariant, we may assume, without losing generality, that A =  a 0 0 b  . (3) If a = b, then, up to a scalar multiplier, A is a unitary matrix. We comment on this case below, in Sec. 3. In what follows, we assume that a = b. To simplify subsequent calculations, we impose two additional restrictions: (1) the eigenvalues a and b of the matrix (3) are assumed to be real (whence A is a Hermitian matrix); ∗ Moscow State University, Moscow, Russia, e-mail: ikramov@cs.msu.su. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 50–56. Original article submitted January 6, 2005. 1072-3374/06/1373-4789 c 2006 Springer Science+Business Media, Inc. 4789