A Procedure for the Diagonalization of Normal Matrices

ABSTRACT

The so-called Jacobi Procedure is extended to the case of normal matrices. A stable iterative procedure is described utilizing plane unitary transformations for such matrices which yield both the characteristic values and their associated vectors. Generally, the technique consists of minimizing at each stage the sum of the squares of the off-diagonal elements of the given matrix; however, there is one case in which this leads to no improvement; i.e. the lowest value for the change is non-negative. In this case, it is shown that a convergent procedure is still possible.

REFERENCES

• 1: C. C J. JAcoBI, tVber ein leichtes Verf~hren, die in der Theorm der Sakularstorungen vorkommenden Gleiehungen numerisch aufzulosen, J Reme Angew. Math. 80, 51-95 (a846).
• 2: R. T. GRE~oRY, Computing eigenvalues and elgenvect, ors, Math. Tables A~ds Comp. 7, 215-220 (1953).
• 3: David A. Pope , C. Tompkins, Maximizing Functions of Rotations—Experiments Concerning Speed of Diagonalization of Symmetric Matrices Using Jacobi's Method, Journal of the ACM (JACM), v.4 n.4, p.459-466, Oct. 1957 [doi>10.1145/320893.320901]
• 4: G. E FORSV'rI~E AND P. HE,x'R~CJ, The cyclic J~cobI reef, hod for computing the principal values of a complex m~trix, to be published.
• 5: A. S. ~OUSEHOLDER, Pmnczptes of Numerzcal Analys~s, pp. 160-162, McGraw-Hill, New York, 1953
• 6: Zv~. LOTKIN, CharacLenstic val(ms of arbitrary mat, rices, Quarl Appl Math 14, 267~275 (1956).
• 7: J. (~REENSTADT, A method for finding roots of arbitrary matrices, Math Tables Aids Comp. 9, 47-52 (t955).
• 8: l{. L. CAUSEY, Computing eigenvalues of non-Hermi~ian matrices by mebhods of Jacobi type, to be published
• 9: J yon Nsu>tA~'N, Mathematical Foundattons of Quantum Mechamcs, pp. 170-178 and pp. 223-229, Pr, nceton University Press, Princeton, 1955
• 10: J. H. M. W~DD~I~BURN, Lectures on Matrices, ch VII, American Mathematical Society Coil Publ 17, New York, 1934
• 11: H. H. Goldstine , F. J. Murray , J. von Neumann, The Jacobi Method for Real Symmetric Matrices, Journal of the ACM (JACM), v.6 n.1, p.59-96, Jan. 1959 [doi>10.1145/320954.320960]

INDEX TERMS

Primary Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.1 Numerical Algorithms and Problems
          Subjects: Computations on matrices

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.3 Numerical Linear Algebra
          Subjects: Eigenvalues and eigenvectors (direct and iterative methods)

General Terms:
Performance, Theory