Proving $\det \left( \begin{smallmatrix} A & -B \\ B & A \end{smallmatrix} \right) =|\det(A+iB)|^2$

Question

The complex general linear group is a subgroup of the group of real matrices of twice the dimension and with positive determinant.

Let us decompose complex matrices $M$ as $M=A+iB$, where $A,B$ are real matrices. Now consider the correspondence $$f(A+iB)=\begin{pmatrix} A & -B \\ B & A\end{pmatrix}.$$

If $\det f(M)=|\det M|^2$ for square matrices, then we would have $GL(n,\mathbb C)\subseteq GL_+(2n,\mathbb R)$ with the identification $M\to f(M)$, which is an injective homomorphism. In other words, the complex general linear group would be a subgroup of the group of real matrices of twice the dimension and with positive determinant.

How is $\det f(M)=|\det M|^2$?

Answer 1

Knowing that the determinant of a real matrix treated as a complex one is the same and that fundamental operations do not change the determinant we get: $$\det \left(\begin{array}{} A & -B\\B & A\end{array}\right)=\det\left(\begin{array}{} A-iB & -B\\B+iA & A\end{array}\right)= \det\left(\begin{array}{} \bar{M} & -B\\i \bar M & A\end{array}\right)= \det\left(\begin{array}{} \bar M & -B\\i \bar M -i \bar M & A +i B\end{array}\right)= \det \left(\begin{array}{} \bar M & -B\\0 & M\end{array}\right)=|\det M|^2$$

Answer 2

More generally, let $\mathbb{L}/\mathbb{K}$ be a finite Galois extension of fields.

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Lemma 0: The (explicit) map $$\mathbb{L}\otimes_{\mathbb{K}}\mathbb{L}\ \overset{\left\langle\left\langle 1_{\mathbb{L}},\ \sigma\right\rangle\right\rangle_{\sigma\ \in\ \text{Gal}\left(\mathbb{L}/\mathbb{K}\right)}}{\longrightarrow}\ \prod_{\sigma\ \in\ \text{Gal}\left(\mathbb{L}/\mathbb{K}\right)}\mathbb{L}$$ is an isomorphism of $\mathbb{K}$-algebras. (Here the outer bracket is the correspondence from the universal property of the product $\prod_{\text{Gal}\left(\mathbb{L}/\mathbb{K}\right)}$ and the inner bracket is the correspondence from the universal product of the pushout $\otimes_{\mathbb{K}}$.)

Proof: Well-known. $\blacksquare$

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Now recall the familiar $\mathbb{K}$-linear functors $$\mathbb{L}\textbf{-vect}\ \overset{\overset{\text{Ind}^{\mathbb{K}}_{\mathbb{L}}}{\longleftarrow}}{\underset{\text{Res}^{\mathbb{L}}_{\mathbb{K}}}{\longrightarrow}}\ \mathbb{K}\textbf{-vect}\text{,}$$ namely restriction and extension of scalars. For every $\sigma\in\text{Gal}\left(\mathbb{L}/\mathbb{K}\right)$ there is moreover a $\mathbb{K}$-linear functor $$\mathbb{L}\textbf{-vect}\ \overset{\text{Res}_{\sigma}}{\longrightarrow}\ \mathbb{L}\textbf{-vect}$$ given by restriction of scalars along $\sigma$.

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Proposition 1: There is a(n explicit) natural isomorphism $$\text{Ind}^{\mathbb{K}}_{\mathbb{L}}\circ\text{Res}^{\mathbb{L}}_{\mathbb{K}}\ \simeq\ \bigoplus_{\sigma\ \in\ \text{Gal}\left(\mathbb{L}/\mathbb{K}\right)} \text{Res}_{\sigma}\text{.}$$

Proof: As $\text{Res}^{\mathbb{L}}_{\mathbb{K}}$, $\text{Ind}^{\mathbb{K}}_{\mathbb{L}}$, each $\text{Res}_{\sigma}$ and thus both $\text{Ind}^{\mathbb{K}}_{\mathbb{L}}\circ\text{Res}^{\mathbb{L}}_{\mathbb{K}}$ and $\bigoplus_{\sigma\ \in\ \text{Gal}\left(\mathbb{L}/\mathbb{K}\right)} \text{Res}_{\sigma}$ are cocontinuous, it suffices by Eilenberg-Watts to check that $$\text{Ind}^{\mathbb{K}}_{\mathbb{L}}\circ\text{Res}^{\mathbb{L}}_{\mathbb{K}}\left(\mathbb{L}\right)\ \simeq\ \bigoplus_{\sigma\ \in\ \text{Gal}\left(\mathbb{L}/\mathbb{K}\right)} \text{Res}_{\sigma}\left(\mathbb{L}\right)$$ as $\mathbb{L}\otimes_{\mathbb{K}}\mathbb{L}$-modules, where the former copy of $\mathbb{L}$ acts "outside the functorial arguent" and the latter copy of $\mathbb{L}$ acts "inside the functorial arguent" (and by $\mathbb{K}$-linearity the two actions coincide on $\mathbb{K}$, as required).

By the construction of $\text{Ind}^{\mathbb{K}}_{\mathbb{L}}$, $$\text{Ind}^{\mathbb{K}}_{\mathbb{L}}\circ\text{Res}^{\mathbb{L}}_{\mathbb{K}}\left(\mathbb{L}\right)\ \simeq\ \mathbb{L}\otimes_{\mathbb{K}}\text{Res}^{\mathbb{L}}_{\mathbb{K}}\left(\mathbb{L}\right)\ \simeq\ \mathbb{L}\otimes_{\mathbb{K}}\mathbb{L}$$ as $\mathbb{L}\otimes_{\mathbb{K}}\mathbb{L}$-modules in the above sense.

At the same time the $\mathbb{L}\otimes_{\mathbb{K}}\mathbb{L}$-module structure on $\bigoplus_{\sigma\ \in\ \text{Gal}\left(\mathbb{L}/\mathbb{K}\right)} \text{Res}_{\sigma}\left(\mathbb{L}\right)$ as above is readily seen to be precisely that on $\prod_{\sigma\ \in\ \text{Gal}\left(\mathbb{L}/\mathbb{K}\right)}\mathbb{L}$ via the map $$\mathbb{L}\otimes_{\mathbb{K}}\mathbb{L}\ \overset{\left\langle\left\langle 1_{\mathbb{L}},\ \sigma\right\rangle\right\rangle_{\sigma\ \in\ \text{Gal}\left(\mathbb{L}/\mathbb{K}\right)}}{\longrightarrow}\ \prod_{\sigma\ \in\ \text{Gal}\left(\mathbb{L}/\mathbb{K}\right)}\mathbb{L}$$ (identifying the biproduct with the product via the canonical isomorphism therebetween).

The claim now follows from Lemma 0. $\blacksquare$

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Finally, denote by $\textbf{B}\omega$ the category presented as having set of objects the singleton, set of morphisms the natural numbers, and composition addition, so that the functor categories $\textbf{B}\omega\to\mathbb{L}\textbf{-vect}$ and $\textbf{B}\omega\to\mathbb{K}\textbf{-vect}$ are the categories of (vector space, endomorphism) pairs over $\mathbb{L}$ and $\mathbb{K}$ respectively. (These are also just the categories of $\mathbb{L}\left[x\right]$- and $\mathbb{K}\left[x\right]$-modules respectively, but the above framing is more conducive to what follows.)

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Corollary 2: There is a(n explicit) natural isomorphism $$\left(1_{\textbf{B}\omega}\to\text{Ind}^{\mathbb{K}}_{\mathbb{L}}\right)\circ\left(1_{\textbf{B}\omega}\to\text{Res}^{\mathbb{L}}_{\mathbb{K}}\right)\ \simeq\ \bigoplus_{\sigma\ \in\ \text{Gal}\left(\mathbb{L}/\mathbb{K}\right)} \left(1_{\textbf{B}\omega}\to\text{Res}_{\sigma}\right)\text{.}$$

Proof: This is immediate from Proposition 1 and that composition and (extant) biproducts are pointwise in functor catgeories, so that $$\left(1_{\textbf{B}\omega}\to\text{Ind}^{\mathbb{K}}_{\mathbb{L}}\right)\circ\left(1_{\textbf{B}\omega}\to\text{Res}^{\mathbb{L}}_{\mathbb{K}}\right)\ \simeq\ \left(1_{\textbf{B}\omega}\to\text{Ind}^{\mathbb{K}}_{\mathbb{L}}\circ\text{Res}^{\mathbb{L}}_{\mathbb{K}}\right)\ \simeq\ \left(1_{\textbf{B}\omega}\to\bigoplus_{\sigma\ \in\ \text{Gal}\left(\mathbb{L}/\mathbb{K}\right)}\text{Res}_{\sigma}\right)\ \simeq\ \bigoplus_{\sigma\ \in\ \text{Gal}\left(\mathbb{L}/\mathbb{K}\right)} \left(1_{\textbf{B}\omega}\to\text{Res}_{\sigma}\right)$$ as claimed. $\blacksquare$

Corollary 3: Given a $\mathbb{L}$-linear endomorphism $A\colon\mathbb{L}^{\text{d}}\to\mathbb{L}^{\text{d}}$ with characteristic polynomial $\text{char}_{\mathbb{L}}\left(A\right)$, its underlying $\mathbb{K}$-linear endomorphism (obtained by restricting scalars) has characteristic polynomial $$\text{char}_{\mathbb{K}}\left(\text{Res}^{\mathbb{L}}_{\mathbb{K}}\left(A\right)\right)\ =\ \prod_{\sigma\ \in\ \text{Gal}\left(\mathbb{L}/\mathbb{K}\right)} \sigma\left(\text{char}_{\mathbb{L}}\left(A\right)\right)\text{.}$$

Proof: This is immediate from Proposition 1, that the characteristic polynomial of a biproduct of matrices is the product of their individual characteristic polyomials, and that characteristic polynomials are preserved (with the appropriate homomorphic change of coefficients, which is in this case injective) under extension of scalars. $\blacksquare$

Corollary 4: Given a $\mathbb{L}$-linear endomorphism $A\colon\mathbb{L}^{\text{d}}\to\mathbb{L}^{\text{d}}$, its underlying $\mathbb{K}$-linear endomorphism (obtained by restricting scalars) has trace $$\text{tr}_{\mathbb{K}}\left(\text{Res}^{\mathbb{L}}_{\mathbb{K}}\left(A\right)\right)\ =\ \sum_{\sigma\ \in\ \text{Gal}\left(\mathbb{L}/\mathbb{K}\right)} \sigma\left(\text{tr}_{\mathbb{L}}\left(A\right)\right)\ :=\ \text{Tr}_{\mathbb{L}/\mathbb{K}}\left(\text{det}_{\mathbb{L}}\left(A\right)\right)$$

Proof: This is immediate from Proposition 1, that the trace of a biproduct of matrices is the sum of their individual traces, and that traces are preserved (with the appropriate homomorphic change of value, which is in this case injective) under extension of scalars. Alternatively this is a direct consquence of Corollary 3. $\blacksquare$

Corollary 5: Given a $\mathbb{L}$-linear endomorphism $A\colon\mathbb{L}^{\text{d}}\to\mathbb{L}^{\text{d}}$, its underlying $\mathbb{K}$-linear endomorphism (obtained by restricting scalars) has determinant $$\text{det}_{\mathbb{K}}\left(\text{Res}^{\mathbb{L}}_{\mathbb{K}}\left(A\right)\right)\ =\ \prod_{\sigma\ \in\ \text{Gal}\left(\mathbb{L}/\mathbb{K}\right)} \sigma\left(\text{det}_{\mathbb{L}}\left(A\right)\right)\ :=\ \text{N}_{\mathbb{L}/\mathbb{K}}\left(\text{det}_{\mathbb{L}}\left(A\right)\right)$$

Proof: This is immediate from Proposition 1, that the determinant of a biproduct of matrices is the product of their individual determinants, and that determinants are preserved (with the appropriate homomorphic change of value, which is in this case injective) under extension of scalars. Alternatively this is a direct consequence of Corollary 3. $\blacksquare$