No. 58 (00174, 00195)

Family name : Robuk
Given name : Victor
Affiliation : Joint Institute for Nuclear Research
Abbreviation : JINR
E-mail address : Victor.Robuk@jinr.ru
Title : A constructive formula for function of a matrix. (Alternative to
        the Lagrange-Silvestre formula).
Authors : V. N. Robuk
Abstract : 
The well-known formula of Lagrange and Silvestre for calculation of
function of finite-dimensional matrix $M$ has certain drawbacks. In
particularly,

i) the arguments of this function are the eigenvalues of the matrix $M$.
In general case of arbitrary matrix it is not possible to obtain the
analytical formula which expresses the eigenvalues in terms of the
elements of the matrix;

ii) accuracy of the numerical calculation cannot be high enough because
it requires approximate calculations, first, of the function arguments,
and second, the function itself;

iii) the Lagrange-Silvestre formula does not apply in general case, in
particular, in case of the degenerate eigenvalues. In the latter
situation the formula requires modifications.

In the present work we suggest an alternative formula for constructing
the function of a matrix. For this purpose we use a hypergeometrical
functions of special type, where arguments are the invariant (trace,
determinant, and others coefficients of characteristic polinomial) of
the original matrix $M$.  Contrary to the eigenvalues, the matrix
invariants can easily be calculated through the elements of the matrix
$M$ by applying the finite number of operations of the addition and the
multiplication.

In short, instead of "elegant" functions (Lagrange-Silvestre formula)
depending on badly-defined arguments (eigenvalue) we suggest
"cumbersome" functions (hypergeometrical series) depending on
well-defined arguments (matrix invariants).

The explicit expressions for the hypergeometrical functions of an
arbitrary finite-dimension matrix are presented.

The function of a matrix $M$ is the sum of any formal finite or infinite
power series of $M$ with arbitrary coefficients.

The obtained results can be used also in analytical solution of the
linear differential equations, and in the group representation theory.
At the same time, if there is need in numerical calculations, the
suggested explicit expressions for the hypergeometrical functions
present the simplest algorithm.

                       References

1. J. H. Wilkinson, THE ALGEBRAIC EIGENVALUE PROBLEM,  Clarendon Press,
Oxford, (1965).

2. F. R. Gantmaher,  THE THEORY OF MATRIX,  Moscow, "NAUKA", Fizmatlit,
(1967), (in Russian).

3. G. H. Golub and Ch. V. Loan, MATRIX COMPUTATIONS, Johns Hopkins
University Press, (1996).