A note on the Hayes-Scott theorem (Q1281210)

Let in the space \(\mathbb{C}^{n}\) an Euclidean scalar product be given, i.e. for vectors \( x= (\alpha_{1}, \ldots, \alpha_{n})^{T}\) and \(y= (\beta_{1}, \ldots, \beta_{n})^{T}\) we have \[ (x, y)= \alpha_{1}\beta_{1} +\ldots + \alpha_{n}\beta_{n}. \] The vector \( x\) is called isotropic if \[ (x, x)= \alpha_{1}^{2} + \ldots +\alpha_{n}^{2}= 0. \] The following is a known theorem [cf. \textit{M. Hayes}, Arch. Ration. Mech. Anal. 85, 41-79 (1984; Zbl 0537.73015)]: Let \(Q\) be a complex symmetric matrix of order 3. An isotropic eigenvector \(x\) of the matrix \(Q\) exists iff \(Q\) has a multiple eigenvalue \(\lambda\). This theorem is connected with a problem of the propagation of waves in an elastic medium. Later on [\textit{N. H. Scott}, Proc. R. Soc. Lond., Ser. A 440, No. 1909, 431-442 (1993; Zbl 0797.15008)] the theorem was generalized to the case of an arbitary dimension \(n\). Here it is shown that the Hayes-Scott theorem is a simple corollary of well known facts of the theory of canonical forms of symmetrical operators in complex Euclidean spaces.