Complex-Linear Matrices & C-Linear Transformations .... Tapp, Propn 2.4 ....

In summary, Peter is reading Kristopher Tapp's book: Matrix Groups for Undergraduates and is currently studying Section 1 in Chapter 2, specifically "1. Complex Matrices as Real Matrices". He needs help understanding how to prove an assertion related to Tapp's Proposition 2.4, which states that a map ##T:V\to W## between ##\mathbb{C}##-vector spaces is ##\mathbb{C}##-linear if and only if if ##\mathbb{R}##-linear and ##T(iv)=iT(v)## for all ##v\in V##. In the remarks following Proposition 2.4, it is stated that ##F## is
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TL;DR Summary
I am reading Kristopher Tapp's book: Matrix Groups for Undergraduates.

I am currently focused on and studying Section 1 in Chapter2, namely:

"1. Complex Matrices as Real Matrices".

I need help in fully understanding how to prove an assertion related to Tapp's Proposition 2.4.
I am reading Kristopher Tapp's book: Matrix Groups for Undergraduates.

I am currently focused on and studying Section 1 in Chapter2, namely:

"1. Complex Matrices as Real Matrices".

I need help in fully understanding how to prove an assertion related to Tapp's Proposition 2.4.

Proposition 2.4 and some comments following it read as follows:

Tapp - Defn 2.3 & Proposition 2.4 ... .png


In the remarks following Proposition 2.4 we read the following:

" ... ... It (##F##) is ##\mathbb{C}##-linear if and only if ##F(i \cdot X) = i \cdot F(X)## for all ##X \in \mathbb{C}^n## ... "My question is as follows ... can someone please demonstrate a proof of the fact that ##F## is ##\mathbb(C)##-linear if and only if ##F(i \cdot X) = i \cdot F(X)## for all ##X \in \mathbb{C}^n## ...
Help will be much appreciated ...

Peter===================================================================================
*** EDIT ***

After a little reflection it appears that " ... ##F## is ##\mathbb{C}##-linear ##\Longrightarrow F(i \cdot X) = i \cdot F(X)## ... " is immediate as ...

... taking ##c = i## we have ...

##F(c \cdot X ) = c \cdot F(X) \Longrightarrow F(i \cdot X) = i \cdot F(X)## for ##c \in \mathbb{C}##Is that correct?

Peter

=======================================================================================
=======================================================================================

Note that Tapp defines##\rho_n## and ##f_n## in the following text ... ...
Tapp - 1 - Chapter 2, Section 1 - PART 1 ... .png

Tapp - 2 - Chapter 2, Section 1 - PART 2 ... .png

Also note that ##R_B## (actually ##R_A##) is defined in the following text ...

Tapp - Defn 1.9 & Defn 1.10 ... .png
Hope the provision of the above text helps with definitions, notation and context ...

Peter
 
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The proof uses that a map ##T:V\to W## between ##\mathbb{C}##-vector spaces is ##\mathbb{C}##-linear if and only if if ##\mathbb{R}##-linear and ##T(iv)=iT(v)## for all ##v\in V##.

From left to right is obvious by the definition, from right to left requires a little easy two-line proof.

Note that in the proof one says that ##T## is ##\mathbb{C}## linear if and only if ##T(iv) = iT(v)## for all ##v\in V## since it was already established that ##T## was ##\mathbb{R}##-linear.
 
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Math_QED said:
The proof uses that a map ##T:V\to W## between ##\mathbb{C}##-vector spaces is ##\mathbb{C}##-linear if and only if if ##\mathbb{R}##-linear and ##T(iv)=iT(v)## for all ##v\in V##.

From left to right is obvious by the definition, from right to left requires a little easy two-line proof.

Note that in the proof one says that ##T## is ##\mathbb{C}## linear if and only if ##T(iv) = iT(v)## for all ##v\in V## since it was already established that ##T## was ##\mathbb{R}##-linear.
Thanks for the post and the hint ...

We are considering ##F \ : \ \mathbb{C}^n \to \mathbb{C}^n## where ##F = f_n^{ -1 } \circ R_B \circ f_n## ...

We know that ##F## is ##\mathbb{R}##-linear ... that is ...

##F(X + Y) = F(X) + F(Y)## for all ##X, Y \in \mathbb{C}^n## ... ... ... ... ... (1)

... and ...

##F(rX) = r F(X)## for all ##X \in \mathbb{C}^n## and ##r \in \mathbb{R}## ... ... ... ... ... (2)We have to prove that ##F## is ##\mathbb{C}##-linear ... ...

... that is ...

##F(X + Y) = F(X) + F(Y)## for all ##X, Y \in \mathbb{C}^n## ...

... which holds from the ##\mathbb{R}##-linear case ...

and ...

##F(cX) = cF(X)## for all ##X \in \mathbb{C}^n## and ##c \in \mathbb{C}## ... ... ... ... ... (3)So ... essentially we have to prove (3) ...

So then ... let ##c = a + bi## ... and proceed as follows ...

##F(cX) = F( (a+bi)X)##

##= F(aX + biX)##

##= F(aX) + F(biX)## since ##F## is ##\mathbb{R}##-linear

##= F(aX) + iF(bX)## since ##F(iY) = iF(Y)##

##= aF(X) + biF(X)## since ##F## is ##\mathbb{R}##-linear

##= (a +bi)F(X)##

##= cF(X)##Hope the above is correct ... ...

Peter
 
  • #4
Math Amateur said:
Thanks for the post and the hint ...

We are considering ##F \ : \ \mathbb{C}^n \to \mathbb{C}^n## where ##F = f_n^{ -1 } \circ R_B \circ f_n## ...

We know that ##F## is ##\mathbb{R}##-linear ... that is ...

##F(X + Y) = F(X) + F(Y)## for all ##X, Y \in \mathbb{C}^n## ... ... ... ... ... (1)

... and ...

##F(rX) = r F(X)## for all ##X \in \mathbb{C}^n## and ##r \in \mathbb{R}## ... ... ... ... ... (2)We have to prove that ##F## is ##\mathbb{C}##-linear ... ...

... that is ...

##F(X + Y) = F(X) + F(Y)## for all ##X, Y \in \mathbb{C}^n## ...

... which holds from the ##\mathbb{R}##-linear case ...

and ...

##F(cX) = cF(X)## for all ##X \in \mathbb{C}^n## and ##c \in \mathbb{C}## ... ... ... ... ... (3)So ... essentially we have to prove (3) ...

So then ... let ##c = a + bi## ... and proceed as follows ...

##F(cX) = F( (a+bi)X)##

##= F(aX + biX)##

##= F(aX) + F(biX)## since ##F## is ##\mathbb{R}##-linear

##= F(aX) + iF(bX)## since ##F(iY) = iF(Y)##

##= aF(X) + biF(X)## since ##F## is ##\mathbb{R}##-linear

##= (a +bi)F(X)##

##= cF(X)##Hope the above is correct ... ...

Peter

Yes, completely correct. Well done!
 
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1. What is a complex-linear matrix?

A complex-linear matrix is a matrix whose entries are complex numbers. This means that the matrix can contain both real and imaginary components.

2. How do you perform complex-linear transformations?

To perform a complex-linear transformation, you can use a complex-linear matrix to multiply with a complex vector. This will result in a new complex vector that has been transformed by the matrix.

3. What is the difference between a complex-linear matrix and a real-linear matrix?

The main difference between a complex-linear matrix and a real-linear matrix is that the entries in a complex-linear matrix can contain imaginary components, while the entries in a real-linear matrix can only contain real numbers.

4. Can complex-linear matrices be used to represent rotations?

Yes, complex-linear matrices can be used to represent rotations in the complex plane. This is because complex numbers can be expressed in polar form, which allows for easy representation of rotations.

5. How are complex-linear matrices used in quantum mechanics?

In quantum mechanics, complex-linear matrices are used to represent operators that act on quantum states. These matrices are known as Hermitian operators and play a crucial role in understanding the behavior of quantum systems.

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