Almost diagonal matrices with multiple or close eigenvalues

If A = D + E where D is the matrix of diagonal elements of A, then when A has some multiple or very close eigenvalues, E has certain characteristic properties. These properties are considered both for hermitian and non-hermitian A. The properties are important in connexion with several algorithms for diagonalizing matrices by similarity transformations.

[1]  C. Jacobi.  Über ein leichtes Verfahren die in der Theorie der Säcularstörungen vorkommenden Gleichungen numerisch aufzulösen*). , 2022 .

[2]  F. L. Bauer.  Optimally scaled matrices , 1963 .

[3]  P. Henrici.  On the Speed of Convergence of Cyclic and Quasicyclic Jacobi Methods for Computing Eigenvalues of Hermitian Matrices , 1958 .

[4]  J. H. Wilkinson.  Note on the quadratic convergence of the cyclic Jacobi process , 1962 .

[5]  H. P. Kempen.  On the convergence of the classical Jacobi method for real symmetric matrices with non-distinct eigenvalues , 1966 .

[6]  David A. Pope,et al.  Maximizing Functions of Rotations—Experiments Concerning Speed of Diagonalization of Symmetric Matrices Using Jacobi's Method , 1957, JACM.

[7]  H. P. Kempen.  On the quadratic convergence of the special cyclic Jacobi method , 1966 .

[8]  James Hardy Wilkinson,et al.  Rigorous Error Bounds for Computer Eigensystems , 1961, Comput. J..

[9]  A. Hoffman,et al.  Lower bounds for the rank and location of the Eigenvalues of a matrix , 1953 .

[10]  A. Hoffman,et al.  The variation of the spectrum of a normal matrix , 1953 .

[11]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[12]  Arnold Schönhage,et al.  Zur quadratischen Konvergenz des Jacobi-Verfahrens , 1964 .

[13]  Arnold Schönhage,et al.  Zur Konvergenz des Jacobi-Verfahrens , 1961 .

[14]  James Hardy Wilkinson.  The QR Algorithm for Real Symmetric Matrices with Multiple Eigenvalues , 1965, Comput. J..

[15]  C. Jacobi,et al.  C. G. J. Jacobi's Gesammelte Werke: Über ein leichtes Verfahren, die in der Theorie der Sacularstorungen vorkommenden Gleichungen numerisch aufzulosen , 1846 .