Asymptotic expansion of the determinant of a perturbed matrix (Q1889698)

A simple and constructive method for obtaining asymptotic expansions of perturbed matrices is presented employing the determinant of the perturbed square \(n \times n\) matrix in the form \[ A(\varepsilon)= \sum_{0}^{\infty} A_k \varepsilon^k, \qquad | \varepsilon| < 1, \] under the assumption that the eigenvalues \(\{\lambda_{0j}\}_1^n\) and the eigenvectors \(\{s_{0j}\}_1^n\) of the nonperturbed (\(\varepsilon=0\)) matrix \(A_0\) are known. This new method allows the simultaneous construction of all eigenvalues \(\lambda_j(\varepsilon)\) and eigenvectors \(s_j(\varepsilon)\), \(j=1, \dots, n\), with the structure of the limit (\(\varepsilon=0\)) matrix \(A_0\) taken into account. The limit matrix \(A_0\) with an arbitrary Jordan structure (with at most one zero eigenvalue) allows to write the asymptotic expansion of the \(\text{ det} \, A(\varepsilon)\) in integer powers of a small parameter \(\varepsilon\) independently of the multiplicity of nonzero points \(\{\lambda_{0j}\}_1^n\) in the spectrum of the matrix \(A_0\).