Complex Derivative: A Basis Free Definition
PARTHA PRATIM DEY
Department of Computer Science
North South University
12 Kemal Ataturk Avenue, Banani, Dhaka
BANGLADESH
Abstract:- Complex variables as we know is very representation dependent, which many think it need not be. Text books define complex numbers asusing basiswhich makes everything including complex derivative, Cauchy-Riemann conditions etc. tied to that particular basis thoughis not the only basis of. In this paper, we use Frechet derivative to construct a definition of complex derivative, which is basis free in the sense that no basis, either explicit or implicit, has been used in the definition.
Key-Words:- Linear operator, matrix, basis, complex, derivative, Frechet derivative.
1 Introduction
Letbe a vector space overwith usual addition and multiplication, specifically
and
where . We define a multiplication between the vectors as follows:
The vetor spaceequipped with the multiplication above is a field denoted by and is called the field of complex numbers. Hence a complex numberis an ordered pair whereandare called real and imaginary parts of and denoted byandrespectively. Notice that we have not used any basis to define our complex number.
Let nowbe a basis for vector spaceand be a -linear operator fromtowith
and
.
For any in, letdenote
,
the transpose of the coordinate matrixrelative to basisand let be the matrix such that theth column of is i.e.,
.
is regarded as a representation of operator and is called the matrix of with respect to basis. For such a matrix
for anyin, as
and
.
Next we quote a well-known theorem [ 1 ] that will be used later to relate matrices of an operator with respect to various bases. The sketch of the proof given below is ours.
TheoremLet be a -linear operator fromto. Ifis a basis ofwithand , then
where
and is the standard basis comprising of .
Proof. Notice that the equations below
is equivalent to the following matrix equation
.
Assume now that and
. Then
.
Similarly one can show that
.
Hence .■
2 Matrix Form of a Multiplication Transformation
Letdenote the - linear transformation from to given by
where and is called a multiplication transformation. The collectionis a subset of and is denoted by. Since
and
the matrix of with respect to the canonical basishas the following form
whereas by Theorem, the matrix of with respect to an arbitrary basisresembles the following
.
Now we prove a theorem, which states given two bases of when the rings of matrices of multiplication transformations are the same.
TheoremLetand denote the rings of matrices of multiplication transformations with respect to basesandrespectively, whereis the canonical basisandis any other. Then =if and only if one of the bases can be obtained from the other by multiplication with a complex number.
Proof.Letandbe two bases ofwhere elements of are obtained by multiplication of each element of by i.e., and. Letandbe representations ofwith respect to basesandrespectively. Thenwhere
.
Since
,
we get
.
.
.
Assumeto be the basiswhere
,
Thenand. Thuscan be rewritten as. Hencebecomes
.
Then
= +
.
We now prove the converse. Letandbe two bases ofand. We want to show thatcan be obtained from by multiplication of each of its element with a complex number, which amounts to one of the following:
a. i.e.,or equivalentlyand
b. i.e.,or equivalentlyand.
Let and be the matrix representations of in basesandrespectively. Thenand hence
.
Since, we obtain = , which implies the following.
,
.
The equalities above are simplified to:
Solving, we get or, which showsis a multiplication ofby a complex number. ■
TheoremLetbe two bases of. Thenif and only if one of the bases can be obtained from the other by multiplication with a complex number.
Proof. Letbe a basis ofand assumebe another where elements ofare obtained by multiplication of each element ofwith an elementi.e.,
Notice that
= .
Then
.
.
.
Hence.
We now prove the converse. Suppose. Then for any, there existsuch that
or equivalently,
.
This implies thatwhereis the canonical basis andcomprises of the transposed first and second columns of. Then by the Theorem, there is a complex numbersuch that,
first column of
and
second column of.
This gives
or
,
which by the discussion in the very beginning of the first part of this proof means the vectors of basiscan be obtained fromby multiplication of each of its elements with complex number.■
3 Matrix Form of Frechet Derivative
Next we discuss a derivative known as the Frechet derivative. Since our definition of complex derivative uses the notion of Frechet derivative, we will say below a few words about Frechet derivative.
Definition. Let be a norm andbe an open set in with respect to this norm. Suppose mapsintoand. If there exists a linear transformationof intosuch that
where
then we say thatis differentiable (or Frechet differentiable) at, and we write
.
If is differentiable at every we say that f is differentiable in.
We consider that maps an open setinto and is differentiable at . Let be the canonical basis of , where as usual and . The components of are real functions ondefined by
.
It is well-known [ 2 ] that
and
.
Thus by Theorem, the Frechet derivative with respect to canonical basis assumes the following form:
.
We will use this important fact to prove our next theorem.
TheoremLetbe a basis foroverandbe a function from an open set towherewith . Ifhas a Frechet derivativeat, then
whereis the matrix of Frechet derivative with respect to basis.
Proof. Set,. Then
and .
Nowcan be rewritten as
whereand constitute the canonical basisof andare as in equationabove. The Frechet derivative of with respect to basisdenoted by is then easily computed by.
using chain rule [2] in the last step.
Setting
,
we thus obtain
.
Since by Theorem (1.1), , we obtain the desired equality. ■
4 A New Definition of Complex Derivative
Now we are ready for a definition of complex derivative.
DefinitionLetbe a map from an open settoandbe it's Frechet derivative on. We say thathas a complex derivative (or is-differentiable) at an elementinif i.e. for some . We letto be .
Recall that in the Introduction, we did not use any basis when we defined the complex numbers. Now notice that our definition of complex derivative does not use a basis either. Nor do we use any specific norm, as the definition of Frechet derivative does not use any. Recall here that all norms are equivalent in a finite dimensional vector space and which is what makes the existence of Frechet derivative independent of any norm.
The next theorem states the necessary and sufficient conditions of complex differentiability of a complex functionat a complex numbergiven an arbitrary basis in.
TheoremSuppose is a basis of over whereand . Letbe a function from an open setto and is given by
when . Thenhas a complex derivativeat if and only if
for some. Moreover.
Proof. Supposeexists. Then for someand . Notice thatimplies . By the last paragraph of Theorem, this amounts to:
.
To get the converse we just go backward. ■
Corollary. ( Cauchy-Riemann Equations) Letdefine a function fromtowith the canonical basiswhereand . Thenhas a complex derivative at if and only if
and
.
Proof. By the theorem above has a complex derivative at if and only if
for some and. Since in this caseisidentity matrix, we obtain
which yields the desired equalities. ■
The corollary above is known in literature as Cauchy-Riemann equations [ 3 ]. These equations are used almost as a definition of -differentiability in textbooks. But notice that they are not basis free, whereas our definition is.
References:
[ 1 ] Howard Anton and Chris Rorres, Elementary Linear Algebra, John Wiley & Sons, Inc., 1994
[ 2 ] Walter Rudin, Principles of Mathematical Analysis, McGraw-Hill International Editions, 1976
[ 3 ] Serge Lang, Complex Analysis, New York: Springer, 1999