The resolvent method and its applications to partial differential equations were developed by Vladimir Dobrushkin (born 1949) in the 1980s. When the resolvent method is applied for defining a function of a square matrix, it is actually based on the Cauchy integral formula where f is holomorphic (meaning that it is represented by a convergent power series) on and inside a closed contour γ that encloses all eigenvalues of a square matrix A. Here j is a unit vector in the positive vertical direction on the complex plane so that \( {\bf j}^2 = -1 . \) When the integrand is a ratio of two polynomials or of entire functions, then the contour integral is the sum of residues (see definition of residue below), which was predicted by a German mathematician Ferdinand Georg Frobenius (1849--1917):
G. Frobenius. Uber die cogredienten Transformationen der bilinearen Formen.
Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 16:7–16, 1896.

For each square n-by-n matrix A, we call a function f(λ) of one complex variable λ admissible for A if f is defined on the spectrum of A (which is the set of all eigenvalues) and all derivatives

exist. Here \( \lambda_1 , \lambda_2 , \ldots , \lambda_m \) are distinct eigenvalues of matrix A of multiplicities \( m_1 , m_2 , \ldots , m_m , \) respectively. ■

A general matrix function is a correspondence that relates to each square matrix A of order n and admissible function f a matrix, denoted by \( f({\bf A}) . \) The matrix \( f({\bf A}) \) has the same order as A with elements in the real or complex field. Such correspondence is assumed to satisfy the following conditions:

• If \( f(\lambda ) = \lambda , \) then \( f({\bf A}) = {\bf A} ; \)
• If \( f(\lambda ) = k, \) a constant, then \( f({\bf A}) = k\,{\bf I} , \) where I is the identity matrix;
• If \( f(\lambda ) = g(\lambda ) + h(\lambda ) , \) then \( f({\bf A}) = g({\bf A}) + h({\bf A}) ; \)
• If \( f(\lambda ) = g(\lambda ) \cdot h(\lambda ) , \) then \( f({\bf A}) = g({\bf A}) \, h({\bf A}) . \)

These requirements will ensure that the definition, when applied to a polynomial p(λ) will yield the usual matrix polynomial p(A) and that any rational identity in scalar functions of a complex variable will be fulfilled by the corresponding matrix functions. The above conditions are not sufficient for most applications, and a fifth requirement would be highly desirable.

• If \( f(\lambda ) = h \left( g(\lambda ) \right) , \) then \( f({\bf A}) = h\left( g({\bf A}) \right) . \)

for holomorphic functions and all admissible functions. The extension of the concept of a function of a complex variable to matrix functions has occupied the attention of a number of mathematicians since 1883. While there are known many approaches to define a function of a square matrix that could be found in the following references:

• R. A. Frazer, W. J. Duncan, and A. R. Collar. Elementary Matrices and Some Applications to Dynamics and Differential Equations. Cambridge University Press, 1938.
• R. F. Rinehart. The equivalence of definitions of a matric function. American Mathematical Monthly, 62, No 6, 395--414, 1955.
• Cleve B. Moler and Charles F. Van Loan. Nineteen dubious ways to compute the exponential of a matrix. SIAM Review, 20, No 4, 801--836, 1978.
• Nicholas J. Higham, Functions of Matrices. Theory and Computation. SIAM, 2008,

we present another method, which is more understandable at undergraduate level.

Recall that the resolvent of a square matrix A is

which is a matrix-function depending on a parameter λ. In general, the resolvent, after reducing all common multiples, is a ratio of a polynomial matrix \( {\bf Q}(\lambda ) \) of degree at most \( k-1 , \) where k is the degree of the minimal polynomial \( \psi (\lambda ): \) Recall that the minimal polynomial for a square matrix A is the unique monic polynomial ψ of lowest degree such that \( \psi ({\bf A} ) = {\bf 0} . \) It is assumed that polynomials \( {\bf Q}(\lambda ) \) and \( \psi (\lambda ) \) are relatively prime (this means that they do not have common multiples). Then, the polynomial in the denominator of the reduced resolvent formula is the minimal polynomial for the matrix A.

The residue of the ratio \( \displaystyle f(\lambda ) = \frac{P(\lambda )}{Q(\lambda )} \) of two polynomials (or entire functions) at the pole \( \lambda_0 \) of multiplicity m is defined by

Recall that a function f(λ) has a pole of multiplicity m at \( \lambda = \lambda_0 \) if, upon multiplication by \( \left( \lambda - \lambda_0 \right)^m , \) the product \( f(\lambda )\, \left( \lambda - \lambda_0 \right)^m \) becomes a holomorphic function in a neighborhood of \( \lambda = \lambda_0 . \) In particular, for simple pole \( m=1 , \) we have For double pole, \( m=2 , \) we have and for triple pole, \( m=3 , \) we get

If a real-valued function \( f(\lambda ) \) has a pair of complex conjugate poles \( a \pm b {\bf j} \) (here \( {\bf j}^2 =-1 \) ), then where Re = \( \Re \) stands for the real part of a complex number. ■

Let \( f(\lambda ) \) be an admissible function defined on the spectrum of a square matrix A. Then

Note that all residues at eigenvalues of A in the above formula exist for admissible functions. We mostly interested for matrix functions corresponding to functions of one variable λ: \( \displaystyle e^{\lambda\,t} , \quad \frac{\sin \left( \sqrt{\lambda}\,t \right)}{\sqrt{\lambda}} , \quad \cos \left( \sqrt{\lambda}\,t \right) \) because they are solutions of the following initial value problems (where dots stand for derivatives with respect to t and I is the identity matrix):

 

To solve another problem with Mathematica, we need to clean the kernel (to do this, click the "Evaluation" fall down button and c hoose "Quit Kernel" option).
Example of functions for a defective matrix.

Example of functions for a defective 3-by-3 matrix.

Example of functions for a complex-valued matrix.

 

Here is the code to calculate a function of a square matrix. The function f= Exp is chosen for simplicity, but it can be any admissible function.

 

For instance, if one wants to calculate cosine of A*t, then type: