When the resolvent method is applied to define a function of a square matrix, it is actually based on the inverse Laplace transform, which can be reduced to the Cauchy integral formula where f is a holomorphic function (meaning that it is represented locally 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 ℂ so that \( {\bf j}^2 = -1 . \) The idea behind defining a function of a linear operator via the Cauchy integral formula came from the French mathematician Jules Henri Poincaré (1854--1912) when he worked on continuous groups in 1899. 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) in 1896. Although our definition of matrix functions is based on previous achievements in this area, it is made understandable for undergraduates by the use of numerous examples.

Our next natural step is to define a set of functions that can be of use for describing matrix functions. It turns out that every matrix uses its own set of functions that we call admissible.

Let f(z) be an admissible function for an n×n matrix A. Extending the definition of f(z) to matrices, we define f(A), formally replacing z by A, yielding another n×n matrix f(A) with entries in the real or complex field. For admissible functions f, g, and h of a matrix A, we require that the following conditions be satisfied:

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

These requirements will ensure that the definition, when applied to a polynomial p(z), 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 conditions above are not sufficient for most applications until a fifth requirement is met:

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

for all admissible functions. The extension of the concept of building a matrix from a given function of a complex variable and a matrix has occupied the attention of a number of mathematicians since 1883. There are many known approaches for defining a function of a square matrix that could be found in the following references [2--5]. Our exposition of the matrix function presents another method, which is based on the residue application to the Cauchy formula.

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 \( \psi (z ): \) Recall that the minimal polynomial of a square matrix A is the unique monic polynomial ψ of lowest degree such that \( \psi ({\bf A} ) = {\bf 0} . \) It is assumed that the n×n matrix polynomial Q(λ) and ψ(λ) are relatively prime (this means that they do not have common multiples containing parameter λ). Then, the polynomial ψ in the denominator of the reduced resolvent formula is the minimal polynomial for the matrix A. Recall that a function f(z) has a pole of multiplicity m at \( z = \lambda_0 \) if, upon multiplication by \( \left( z - \lambda_0 \right)^m , \) the product \( f(z )\, \left( z - \lambda_0 \right)^m \) becomes a holomorphic function in a neighborhood of \( z = \lambda_0 . \) 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 \( f(z ) \) has a pair of complex conjugate poles \( a \pm b {\bf j} \) (here \( {\bf j}^2 =-1 \) ), then where Re z = \( \Re\,z \) stands for the real part of a complex number z.    ■ Note that all residues at eigenvalues of A in the formula above exist for admissible functions. We are mostly interested in matrix functions corresponding to functions of one variable z and depending on a real parameter t associated with time, especially \( \displaystyle e^{z\,t} , \quad \frac{\sin \left( \sqrt{z}\,t \right)}{\sqrt{z}} , \quad \cos \left( \sqrt{z}\,t \right) \) 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):

When a square matrix A is diagonalizable, the function ψ(λ) in formula \eqref{EqResolvent.2} is a product of simple terms, ψ(λ) = (λ - λ₁) ··· (λ - λ_m). In this case, the residues of the resolvent become Sylvester's auxiliary matrices:

where s is an eigenvalue of matrix A. The derivative of the minimal polynomial ψ(λ) evaluated at λ = s can be evaluated without differentiation by dropping a particular factor.

When you solve another problem with Mathematica, don't forget to clean the kernel (to do this, click the "Evaluation" fall down button and choose "Quit Kernel" option). Otherwise the variables in use will inherit previously used values.

 

Moreover, we will show that the two functions, Φ(t) and Ψ(t), are real and imaginary parts of the exponential function:

where Re z = ℜ z and Im z = ℑ z are real and imaginary parts of a complex number z, respectively. Therefore, we obtain Euler's formula for matrices These matrix-functions are solutions of the initial value problems \eqref{EqResolvent.10} --\eqref{EqResolvent.12}, where dots represent differentiation with respect to time variable t: \( \dot{\bf U} = {\text d}{\bf U}/{\text d}t \) and \( \ddot{\bf U} = {\text d}^2 {\bf U}/{\text d}t^2 . \)

Note that not every function that you studied in calculus is suitable for calculation of a corresponding matrix function. For example, a square root function \( f(\lambda ) = \lambda^{1/2} = \sqrt{\lambda} \) is an analytic function, but not a holomorphic function at the origin λ = 0. Therefore, if matrix A is singular, then the corresponding square root matrix function A^1/2 requires residue evaluation at λ = 0, which does not exist. Nevertheless, we demonstrate via some examples that that this square root function can be determined formally when a corresponding matrix A has a minimal polynomial with simple zero. However, if the minimal polynomial has a double of higher zero root (so it contains a multiple λ^m with m ≥ 2), then the root A^1/2 does not exist. For instance, matrix \( {\bf A} = \begin{bmatrix} 0&1 \\ 0&0 \end{bmatrix} \) has minimal polynomial ψ(λ) = λ², and it has no square root. So there does not exist a matrix R such that R² = A. Example 1: 3-by-3 matrix with three positive distinct real eigenvalues

Example 2: 3-by-3 matrix with two positive distinct real eigenvalues

Example 3 of functions for two complex-valued 2×2 matrices Example 4: 3-by-3 matrix with two complex conjugate eigenvalues Example 5: matrix functions for a defective 3×3 matrix

Example 6: matrix functions for a defective nonsingular 3-by-3 matrix

 

We outline all steps needed to define a matrix-function in the folloing example.

 

Example 7: matrix functions for a defective nonsingular 3-by-3 matrix