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Derivative and higher-order Cauchy integral formula of matrix functions

  • Xiaojie Huang EMAIL logo , Zhixiu Liu and Chun Wu
Published/Copyright: December 31, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

The derivative of a n -order matrix function on the complex field is usually defined as a n 2 -order matrix, which is not suitable for generalizing Cauchy integral formula of matrix functions to its higher-order derivative form. In this paper, a new kind of derivative of matrix functions is defined, and the higher-order derivative form of Cauchy integral formula of matrix functions is also proved to be true under the new kind of definition of derivative. At the same time, some examples about calculating the values of matrix functions by using Cauchy integral formula of matrix functions and its higher-order derivative form are given.

Keywords: derivative of matrix functions; Cauchy integral formula; matrix function; Hamilton-Cayley theorem; analytic function
MSC 2010: 15A16; 15A15

1 Introduction and questions

Let C denote the complex field and C n × n denote the matrix ring on C . If f ( z ) is a polynomial function on C and A is a n -order matrix in C n × n , then the meaning of matrix function f : A ∈ C n × n → f ( A ) ∈ C n × n , abbreviated as f ( A ) , is obvious, that is a polynomial whose variable is the matrix A (see [1]). However, for more general function f ( z ) , the meaning of the corresponding matrix function f ( A ) is not so obvious, and how to define it reasonably is an important problem. In fact, not only matrix ring C n × n but also the general Banach algebra on complex field C has the problem about reasonable definition of function f ( A ) . Whether matrix rings or more general Banach algebras, there is always a good scheme to the definition of f ( A ) for analytical function f ( z ) on complex field C . If f ( z ) is an analytic function on complex field C , then it can be expanded by power series f ( z ) = ∑ k = 0 k = ∞ z k , and naturally, the f ( A ) is defined as f ( A ) ≔ ∑ k = 0 k = ∞ A k (see [1,2]).

Although the aforementioned definition of f ( A ) is intuitive, it has some inconveniences. For example, even for matrix A in matrix ring C n × n , not to mention the element in general Banach algebra on C , it is difficult to calculate the value of f ( A ) . Fortunately, on the basis of annihilation polynomial of matrix and Lagrange-Sylvester theorem, we can give the polynomial matrix function equivalent to f ( A ) , which is convenient to calculate the value of f ( A ) (see [1]).

Is there any other equivalent definition of the matrix functions corresponding to analytic functions on the complex field C ? This is a natural problem because the analytic functions have many different equivalent definitions and representations. In fact, in addition to the power series representation, the analytical function f ( z ) on C has many other important characteristics, and it even could be expressed by G derivative only according to the function values on a sequence of points with at least one aggregation point (see [3]), which is a profound supplement to the uniqueness theorem of analytic functions. Of course, analytical function f ( z ) can also be expressed by the function values on a curve, that is the following famous Cauchy integral formula (see [4]).

Theorem 1.1

Let D be the interior domain of rectifiable curve L in C , f ( z ) be an analytical function on D and continuous on closed region D ¯ , then for any z ∈ D ,

f ( z ) = 1 2 π i ∫ L f ( ξ ) ξ − z d ξ .

On complex field C , is there representation of Cauchy integral formula for the function f ( A ) in matrix rings or more general Banach algebras? The answer is affirmative, we have the similar Cauchy integral formula for f ( A ) (see [2]), and more details are in the second part of the paper. In order to be more specific and more relevant with the paper’s topic, we only discuss the matrix function f ( A ) in C n × n . The discussion on Banach algebras may be another topic to be studied.

Furthermore, the Cauchy integral formula of analytical functions has the form of higher-order derivative as follows (see [4]).

Theorem 1.2

Let D be the interior domain of rectifiable curve L in C , f ( z ) be an analytical function on D and continuous on closed region D ¯ , then for any z ∈ D ,

f ( α ) ( z ) = α ! 2 π i ∫ L f ( ξ ) ( ξ − z ) α + 1 d ξ ,

where α is a positive integer, f ( α ) ( z ) is the α th order derivative of f ( z ) .

So do the matrix functions have the corresponding higher-order derivative form? The first problem it has to deal with is how to define the derivative of matrix functions. According to the usual definition, the derivative of

f : A = { A k l } k , l = 1 n ∈ C n × n → f ( A ) = { f ( A ) s t } s , t = 1 n ∈ C n × n ,

where A k l and f ( A ) s t are the elements of matrix A and f ( A ) , respectively, k , l , s , and t are positive integers, is

f ′ ( A ) = d f ( A ) d A = d f ( A ) d A k l k , l = 1 n ,

which is a n 2 -order matrix, where d f ( A ) d A k l = d f ( A ) s t d A k l s , t = 1 n (see [5]). However, the corresponding integral representation

f ′ ( A ) = d f ( A ) d A = 1 2 π i ∫ L f ( ξ ) ( ξ I − A ) 2 d ξ

is a matrix of n -order (if so more details are in the second part of the paper). The aforementioned two matrices are obviously not equivalent. In order to let the matrix functions have Cauchy integral formula in the form of corresponding higher-order derivative, another new and appropriate definition of derivative of matrix functions must be given.

2 Definitions and main results

The square matrix function f ( A ) is defined as follows.

Definition 2.1

[1] Let f ( z ) = ∑ k = 0 ∞ a k z k be an analytic function on domain D = { z : ∣ z ∣ < R } in C , A be a n -order matrix in C n × n , the matrix spectral norm is denoted as ‖ A ‖ and ‖ A ‖ < R and then the matrix series ∑ k = 0 ∞ a k A k converges and be called the matrix function f ( A ) , that is, f ( A ) ≔ ∑ k = 0 ∞ a k A k .

When f ( z ) is analytic in C , the ‖ A ‖ < + ∞ ( = R ) always holds, which means that f ( A ) is always reasonable. As previously mentioned, in order to extend the Cauchy integral formula to the form of higher-order derivative, we need the concept of derivative of matrix functions and define it as follows.

Definition 2.2

Let f ( z ) be an analytic function on domain D = { z : ∣ z ∣ < R } , A be a n -order matrix with spectral norm ‖ A ‖ < R , f ( A ) be the corresponding matrix function, the derivative of order α of f ( z ) is f ( α ) ( z ) and then the matrix function determined by f ( α ) ( z ) is denoted as f ( α ) ( A ) and called the derivative of order α of f ( A ) , where α is a positive integer.

Before stating Cauchy integral formula and its higher-order derivative form of matrix functions, it needs the definition of integral on a curve L in C of matrix valued functions with complex variable. The following definition does not depend on measure (see [6]), but only on algebraic operation, which is similar to Riemann integral (see [5]).

Definition 2.3

Let A ( z ) : z ∈ D → A ( z ) ∈ C n × n , abbreviated as A ( z ) , be a matrix valued function with complex variable, where D is a domain in C . The rectifiable curve L is contained in the domain D , Δ = { z 0 (beginning point), z 1 , z 2 , … , z m ( ending point ) } denotes the partition of the curve L , that is, L = ⋃ i = 0 i = m − 1 z i z i + 1 ^ . ξ i is an arbitrary given point on arc z i z i + 1 ^ , ∣ Δ ∣ ≔ max 1 ≤ i ≤ m − 1 { ∣ z i + 1 − z i ∣ } , if the limit lim ∣ Δ ∣ → 0 ∑ i = 0 m − 1 A ( ξ i ) ( z i + 1 − z i ) exists, it is defined as the integral of matrix valued function A ( z ) on the curve L and denoted as ∫ L A ( z ) d z .

The Cauchy integral formula of matrix function f ( A ) can be obtained as follows.

Theorem 2.4

[1] Let A be a n -order matrix in C n × n and ‖ A ‖ be the matrix spectral norm. The function f ( z ) used to define the matrix function f ( A ) is an analytic function on C . L is a circle centered at the origin of C with the radius r > ‖ A ‖ , then

f ( A ) = 1 2 π i ∫ L f ( ξ ) ξ I − A d ξ .

Based on the aforementioned definition of derivative of matrix functions, we can extend the Cauchy integral formula of matrix functions to the form of higher-order derivative.

Theorem 2.5

Let A be a n -order matrix in C n × n and ‖ A ‖ be the matrix spectral norm. The function f ( z ) used to define the matrix function f ( A ) is an analytic function on C . L is a circle centered at the origin of C with the radius r > ‖ A ‖ , then

f ( α ) ( A ) = α ! 2 π i ∫ L f ( ξ ) ( ξ I − A ) ( α + 1 ) d ξ ,

where α is a positive integer and f ( α ) ( A ) is the α th order derivative of f ( A ) .

Denote A ( z ) as { a k l ( z ) } k , l = 1 n , then ∫ L A ( z ) d z can be denoted as ∫ L a k l ( z ) d z k , l = 1 n . This kind of expression sometimes can be convenient. For this purpose, we synthesize Theorems 2.4 and 2.5 and rewrite them as follows.

Theorem 2.6

Let A = { a k l } k , l = 1 n be a n -order matrix in C n × n and ‖ A ‖ be the matrix spectral norm. The function f ( z ) used to define the matrix function f ( A ) is an analytic function on C . L is a circle centered at the origin of C with the radius r > ‖ A ‖ , and then, ( ξ I − A ) − ( α + 1 ) may be represented with { b k l ( ξ ) } k , l = 1 n and

f ( α ) ( A ) = α ! 2 π i ∫ L f ( ξ ) b k l ( ξ ) d ξ k , l = 1 n ,

where α is a nonnegative integer and f ( α ) ( A ) is the α th order derivative of f ( A ) and f ( 0 ) ( A ) ≔ f ( A ) , 0 ! ≔ 1 .

Remark 2.7

The calculation of ( ξ I − A ) − ( α + 1 ) = { b k l ( ξ ) } k , l = 1 n can be done by elementary transformation (see [7]), which is the key to the application of Theorem 2.6. The integral ∫ L f ( ξ ) b k l ( ξ ) d ξ can be calculated by residue theorem of complex variable functions (see [4]). Calculation examples are in the fourth part of the paper.

3 Proof of the main results

In order to prove Theorem 2.5, we first present the following lemmas. For convenience, we omit the superscript and subscript of curly brackets representing the matrix below.

Lemma 3.1

[4] Let f ( z ) be an analytic function on circle domain D = { z : ∣ z − a ∣ < R } in C , then f ( z ) = ∑ k = 0 ∞ f ( k ) ( a ) k ! ( z − a ) k , ∀ z ∈ D .

Lemma 3.2

Let L be a rectifiable curve and f ( z ) be a continuous function on the curve L , A = { a k l } ∈ C n × n , then ∫ L f ( z ) d z A m = ∫ L f ( z ) A m d z , where m is a nonnegative integer and A 0 ≔ I is an identity matrix.

Proof

Note that ∫ L f ( z ) d z exists, denote A m = { a k l ( m ) } , then

∫ L f ( z ) d z A m = ∫ L f ( z ) d z a k l ( m ) = ∫ L f ( z ) a k l ( m ) d z = ∫ L { f ( z ) a k l ( m ) } d z = ∫ L f ( z ) { a k l ( m ) } d z = ∫ L f ( z ) A m d z . □

Lemma 3.3

[4] Let L be a rectifiable curve and f k ( z ) be continuous functions on the curve L , where k is a positive integer. If series ∑ k = 1 ∞ f k ( z ) uniformly converges to function f ( z ) on L , then ∫ L f ( z ) d z = ∑ k = 1 ∞ ∫ L f k ( z ) d z .

Lemma 3.4

Let A m ( z ) and A ( z ) be n -order matrix valued functions with complex variable, where m is a positive integer, L be a rectifiable curve in C . In the sense of matrix norm, A m ( z ) is continuous on L and series ∑ m = 1 ∞ A m ( z ) uniformly converges to function A ( z ) on L , then

∫ L A ( z ) d z = ∑ m = 1 ∞ ∫ L A m ( z ) d z .

Proof

Denote ∑ m = 1 d A m ( z ) = ∑ m = 1 d { a k l ( m ) ( z ) } , A ( z ) = { a k l ( z ) } , series ∑ m = 1 ∞ A m ( z ) converges uniformly to function A ( z ) on L , then for any given ε > 0 , exists N > 0 , if d > N , for arbitrary z ∈ L ,

∑ m = 1 d A m ( z ) − A ( z ) = ∑ k , l = 1 n ∑ m = 1 d a k l ( m ) ( z ) − a k l ( z ) 2 1 2 < ε ,

and then, the series ∑ m = 1 ∞ a k l ( m ) ( z ) converges uniformly to a k l ( z ) .

By Lemma 3.3, we have ∫ L a k l ( z ) d z = ∑ m = 1 ∞ ∫ L a k l ( m ) ( z ) d z , then,

∑ m = 1 ∞ ∫ L A m ( z ) d z = ∑ m = 1 ∞ ∫ L a k l ( m ) ( z ) d z = ∑ m = 1 ∞ ∫ L a k l ( m ) ( z ) d z = ∫ L a k l ( z ) d z = ∫ L A ( z ) d z .

Although the proof is based on 2-norm, it is not without generality according to the equivalence of norms. The result still holds for the other norm, e.g., spectral norm.□

Lemma 3.5

[7] Let A be n -order matrix in C n × n and ‖ A ‖ be matrix spectral norm and ‖ A ‖ < 1 , then

( I + A ) α = ∑ k = 0 ∞ α k A k ,

where α is an integer, k is a positive integer and α k = α ( α − 1 ) ⋯ ( α − k + 1 ) k ! .

Now we turn to the proof of Theorem 2.5.

Proof

As f ( z ) is analytical in the whole complex field C , by Lemma 3.1,

f ( z ) = ∑ k = 0 ∞ f ( k ) ( 0 ) k ! z k , f ( α ) ( z ) = ∑ k = α ∞ f ( k ) ( 0 ) ( k − α ) ! z k − α ,

according to Definition 2.2,

f ( α ) ( A ) = ∑ k = α ∞ f ( k ) ( 0 ) ( k − α ) ! A k − α .

As a result of Theorem 1.2,

f ( k ) ( 0 ) = k ! 2 π i ∫ L f ( ξ ) ξ k + 1 d ξ ,

then

f ( α ) ( A ) = ∑ k = α ∞ 1 ( k − α ) ! k ! 2 π i ∫ L f ( ξ ) ξ k + 1 d ξ A k − α .

By Lemma 3.2, we have

f ( α ) ( A ) = ∑ k = α ∞ ∫ L 1 ( k − α ) ! k ! 2 π i f ( ξ ) ξ k + 1 d ξ A k − α = ∑ k = α ∞ ∫ L f ( ξ ) k ! ( k − α ) ! 1 2 π i A k − α ξ k + 1 d ξ .

On the other hand, since ( c z ) n = c n z n , by Definition 2.1, ( c A ) n = c n A n . Let the radius of L be large enough such that A ξ < 1 when ξ ∈ L , and according to Lemma 3.5,

( ξ I − A ) − ( α + 1 ) = ξ − ( α + 1 ) I − A ξ − ( α + 1 ) = ξ − ( α + 1 ) ∑ k = 0 ∞ − ( α + 1 ) k − A ξ k = ξ − ( α + 1 ) ∑ k = 0 ∞ ( α + 1 ) ( α + 2 ) ⋯ ( α + k ) k ! A ξ k = ∑ k = 0 ∞ ( α + k ) ! k ! α ! A k ξ α + k + 1 .

So we have

∑ k = α ∞ f ( ξ ) k ! ( k − α ) ! 1 2 π i A k − α ξ k + 1 = α ! 2 π i f ( ξ ) ∑ k = α ∞ k ! ( k − α ) ! α ! A k − α ξ k + 1 = α ! 2 π i f ( ξ ) ∑ k = 0 ∞ ( α + k ) ! k ! α ! A k ξ α + k + 1 = α ! 2 π i f ( ξ ) ( ξ I − A ) − ( α + 1 ) .

By Lemma 3.4,

f ( α ) ( A ) = ∑ k = α ∞ ∫ L f ( ξ ) k ! ( k − α ) ! 1 2 π i A k − α ξ k + 1 d ξ = ∫ L ∑ k = α ∞ f ( ξ ) k ! ( k − α ) ! 1 2 π i A k − α ξ k + 1 d ξ = α ! 2 π i ∫ L f ( ξ ) ( ξ I − A ) − ( α + 1 ) d ξ . □

4 Examples of application

Example 4.1

If A = { a k l } ∈ C n × n , p ( z ) = ∣ z I − A ∣ , then p ( A ) = O .

Proof

Note that { b k l ( ξ ) } ≔ ( ξ I − A ) − 1 = M k l ( ξ ) ∣ ξ I − A ∣ , where M k l ( ξ ) is an algebraic cofactor, which is a polynomial of ξ . By residue theorem or Cauchy integral theorem (see [4]), we have

∫ L M k l ( ξ ) d ξ = 0 .

By Theorem 2.6 and p ( ξ ) = ∣ ξ I − A ∣ ,

p ( A ) = 1 2 π i ∫ L p ( ξ ) M k l ( ξ ) ∣ ξ I − A ∣ d ξ = 1 2 π i ∫ L M k l ( ξ ) d ξ = { 0 } = O . □

Remark 4.2

Example 4.1 is the proof of Hamilton-Cayley theorem on the complex field (see [8]).

Example 4.3

Let A = − 1 1 0 0 − 2 0 0 0 − 3 , calculate 2 A sin ( A ) + A 2 cos ( A ) .

Proof

By using elementary transformation, we can get

{ b k l ( ξ ) } ≔ ( ξ I − A ) − 1 = 1 ξ + 1 1 ( ξ + 1 ) ( ξ + 2 ) 0 0 1 ξ + 2 0 0 0 1 ξ + 3 ,

then

{ b ¯ i j ( ξ ) } ≔ ( ξ I − A ) − 2 = ( ( ξ I − A ) − 1 ) 2 = 1 ( ξ + 1 ) 2 2 ξ + 3 ( ξ + 1 ) 2 ( ξ + 2 ) 2 0 0 1 ( ξ + 2 ) 2 0 0 0 1 ( ξ + 3 ) 2 .

Note that [ z 2 sin ( z ) ] ′ = 2 z sin ( z ) + z 2 cos ( z ) , by Theorem 2.6, then

2 A sin ( A ) + A 2 cos ( A ) = [ A 2 sin ( A ) ] ′ = 1 2 π i ∫ L ( ξ 2 ) sin ( ξ ) b ¯ i j ( ξ ) d ξ = 1 2 π i ∫ L ξ 2 sin ( ξ ) ( ξ + 1 ) 2 d ξ 1 2 π i ∫ L ξ 2 sin ( ξ ) ( 2 ξ + 3 ) ( ξ + 1 ) 2 ( ξ + 2 ) 2 d ξ 0 0 1 2 π i ∫ L ξ 2 sin ( ξ ) ( ξ + 2 ) 2 d ξ 0 0 0 1 2 π i ∫ L ξ 2 sin ( ξ ) ( ξ + 3 ) 2 d ξ .

By using the residue theorem to calculate all the aforementioned integrals,

2 A sin ( A ) + A 2 cos ( A ) = 2 sin ( 1 ) + cos ( − 1 ) 2 sin ( 1 ) + cos ( − 1 ) − 4 sin ( 2 ) − 4 cos ( − 2 ) 0 0 4 sin ( 2 ) + 4 cos ( − 2 ) 0 0 0 6 sin ( 3 ) + 9 cos ( − 3 ) . □

Example 4.4

Let A = − 3 1 2 − 2 and t be a real number, calculate exp ( A t ) .

Proof

By using elementary transformation, we can get

{ b i j ( ξ , t ) } ≔ ( ξ I − A t ) − 1 = ξ + 2 t ( ξ + 4 t ) ( ξ + t ) t ( ξ + 4 t ) ( ξ + t ) 2 t ( ξ + 4 t ) ( ξ + t ) ξ + 3 t ( ξ + 4 t ) ( ξ + t ) ,

and then,

{ b ¯ i j ( ξ , t ) } ≔ ( ξ I − A ) − 2 = ( ξ + 2 t ) 2 + 2 t 2 ( ξ + 4 t ) 2 ( ξ + t ) 2 ( ξ + 2 t ) t + t ( ξ + 3 t ) ( ξ + 4 t ) 2 ( ξ + t ) 2 2 t ( ξ + 2 t ) + 2 t ( ξ + 3 t ) ( ξ + 4 t ) 2 ( ξ + t ) 2 2 t 2 + ( ξ + 3 t ) 2 ( ξ + 4 t ) 2 ( ξ + t ) 2 .

Let f ( z ) = exp ( z ) , which is identical to its derivative. By Theorem 2.6,

exp ( A t ) = f ( A t ) = 1 2 π i ∫ L exp ( ξ ) b i j ( ξ , t ) d ξ = 1 2 π i ∫ L exp ( ξ ) ( ξ + 2 t ) ( ξ + 4 t ) ( ξ + t ) d ξ 1 2 π i ∫ L exp ( ξ ) t ( ξ + 4 t ) ( ξ + t ) d ξ 1 2 π i ∫ L exp ( ξ ) 2 t ( ξ + 4 t ) ( ξ + t ) d ξ 1 2 π i ∫ L exp ( ξ ) ( ξ + 3 t ) ( ξ + 4 t ) ( ξ + t ) d ξ ,

and at the same time,

exp ( A t ) = f ′ ( A t ) = 1 2 π i ∫ L exp ( ξ ) b ¯ i j ( ξ , t ) d ξ = 1 2 π i ∫ L exp ( ξ ) ( ξ + 2 t ) 2 + 2 t 2 ( ξ + 4 t ) 2 ( ξ + t ) 2 d ξ 1 2 π i ∫ L exp ( ξ ) ( ξ + 2 t ) t + t ( ξ + 3 t ) ( ξ + 4 t ) 2 ( ξ + t ) 2 d ξ 1 2 π i ∫ L exp ( ξ ) 2 t ( ξ + 2 t ) + 2 t ( ξ + 3 t ) ( ξ + 4 t ) 2 ( ξ + t ) 2 d ξ 1 2 π i ∫ L exp ( ξ ) 2 t 2 + ( ξ + 3 t ) 2 ( ξ + 4 t ) 2 ( ξ + t ) 2 d ξ .

By using the residue theorem to calculate all the aforementioned integrals,

exp ( A t ) = f ( A t ) = f ′ ( A t ) = exp ( − t ) + 2 exp ( − 4 t ) 3 exp ( − t ) − exp ( − 4 t ) 3 2 exp ( − t ) + 4 exp ( − 4 t ) 3 2 exp ( − t ) + exp ( − 4 t ) 3 . □

Remark 4.5

It is sometimes more convenient to calculate the value of matrix function by Theorem 2.6 and residue theorem than by the annihilation polynomial of matrix and Lagrange-Sylvester theorem.

Acknowledgements

This work was supported in part by Science and Technology Project of Jiangxi Provincial Department of Education (Nos. GJJ180944 and GJJ190963) and Chongqing Natural Science Foundation Project (No. cstc2019jcyj-msxmX0390).

  1. Conflict of interest: The authors state no conflict of interest.

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Received: 2021-09-17
Revised: 2021-12-07
Accepted: 2021-12-08
Published Online: 2021-12-31

© 2021 Xiaojie Huang et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  74. On reducible non-Weierstrass semigroups
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  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
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  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
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  83. The role of w-tilting modules in relative Gorenstein (co)homology
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  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
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  113. Solutions to a multi-phase model of sea ice growth
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  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
https://doi.org/10.1515/math-2021-0135
Keywords for this article
derivative of matrix functions; Cauchy integral formula; matrix function; Hamilton-Cayley theorem; analytic function
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