I Erroneously finding discrepancy in transpose rule

[nomadreid]

TL;DR

The rule for switching rows and columns to form the transpose of a matrix seems to come up with two different results for a+bi, considered as a scalar or as a matrix. What is my error?

Obviously, there is something elementary I am missing here.

To form the transpose of a matrix, one exchanges rows and columns, so the transpose of a scalar, considered as (or isomorphic to) a one-entry matrix, should stay the same, including if the scalar is a complex number. On the other hand, in the isomorphism between the complex plane and the real plane, a complex number a+bi corresponds to a matrix

in the real plane; taking the transpose we get

which then corresponds to a-bi back in the complex plane.

I am making some elementary error here. What?

 

[fresh_42]

$\mathbb{C}\not\cong\mathbb{R}^2$

The complex numbers are a field; the complex plane is not a field. If you want to identify them, or, as in this case, with a subalgebra of $\mathbb{R}^4,$ you have to decide which properties you want to preserve in this identification. Transposition here becomes complex conjugation.

 

[nomadreid]

Interesting! Thank you very much, fresh_42. I will look further into that.

 

[martinbn]

TL;DR: The rule for switching rows and columns to form the transpose of a matrix seems to come up with two different results for a+bi, considered as a scalar or as a matrix. What is my error?

Obviously, there is something elementary I am missing here.

To form the transpose of a matrix, one exchanges rows and columns, so the transpose of a scalar, considered as (or isomorphic to) a one-entry matrix, should stay the same, including if the scalar is a complex number. On the other hand, in the isomorphism between the complex plane and the real plane, a complex number a+bi corresponds to a matrix
View attachment 367772
in the real plane; taking the transpose we get
View attachment 367773
which then corresponds to a-bi back in the complex plane.

I am making some elementary error here. What?

What do you mean by a matrix in the real plane? The real plane is not the set of such matrices. More importantly why do you expect this isomorphism (between the complex numbers and the set of such matrices) to respect transpose?

 

[nomadreid]

Thanks very much, martinbn. As fresh_42 pointed out, I was (erroneously) thinking of the real plane along with extra structure, i.e., perhaps as a part of ℝ⁴, and also as fresh_42 pointed out, there isn't the isomorphism which secondary school introductions to the complex plane give the impression of existing, and any correspondence is going to lose some information (such as transpose).

 

[martinbn]

Thanks very much, martinbn. As fresh_42 pointed out, I was (erroneously) thinking of the real plane along with extra structure, i.e., perhaps as a part of ℝ⁴, and also as fresh_42 pointed out, there isn't the isomorphism which secondary school introductions to the complex plane give the impression of existing, and any correspondence is going to lose some information (such as transpose).

But the usual identification of the complex numbers and the real plane sends $a+bi$ to $(a,b)$. You are looking at something different.

 

[nomadreid]

Again, thanks, martinbn. Indeed. I think you mean I oversimplified or neglected the structures that accompany each.