Matrix Rings and the Adjoint

Introduction

When the endomorphism ring of Involutions of the Endomorphism Ring is a matrix ring, the adjoint becomes a concrete formula: with respect to a sesquilinear form with Gram matrix $\Phi$ and coefficient involution $\sigma$, the adjoint of a matrix $X$ is $\Phi^{-1}\sigma(X)^{\mathrm t}\Phi$, and with respect to the $\sigma$-twisted trace pairing of the category it is read off from the left and the right multiplications. Three objects come together in the matrix case and are easy to confuse: the adjoint involution $X \mapsto \Phi^{-1}\sigma(X)^{\mathrm t}\Phi$ defined by a form, the self-adjoint and skew-adjoint matrices it picks out, and the unitary group of the matrices with $X^{*}X = XX^{*} = I$, which are exactly the matrices preserving the form. This article computes the adjoint in the matrix ring, identifies the self-adjoint matrices and the unitary group, and relates them to the trace pairing of the ring.

It assumes Matrix Rings with an Involution, Involutions of the Endomorphism Ring for the adjoint, and The Transpose as an Adjoint for the trace duality; the sesquilinear forms themselves and their classification are Part II, and the Gram matrix is used only as a matrix. Throughout, $R$ is a commutative ring with $1 \neq 0$ and an involution $\sigma$, $\Phi \in M_n(R)$ is a matrix with $\Phi$ invertible when it is used as a Gram matrix, $\tau$ is the matrix trace, and $X^{*} = \Phi^{-1}\sigma(X)^{\mathrm t}\Phi$.

The Adjoint Defined by a Form

Theorem. Let $\Phi$ be invertible and let the sesquilinear form be $\langle x,y\rangle = \sigma(x)^{\mathrm t}\Phi y$ on $R^n$. Then

$$ \langle Xx,y\rangle = \langle x, X^{*}y\rangle , \qquad X^{*} = \Phi^{-1}\sigma(X)^{\mathrm t}\Phi , $$

so $X\mapsto X^{*}$ is the adjoint involution of $M_n(R)$ determined by $\Phi$ and $\sigma$; for $\Phi = I$ it is $X\mapsto \sigma(X)^{\mathrm t}$, and for $\sigma = \mathrm{id}$ and $\Phi = I$ it is the transpose.

Proof. $\langle Xx,y\rangle = \sigma(Xx)^{\mathrm t}\Phi y = \sigma(x)^{\mathrm t}\sigma(X)^{\mathrm t}\Phi y = \sigma(x)^{\mathrm t}\Phi(\Phi^{-1}\sigma(X)^{\mathrm t}\Phi)y = \langle x, X^{*}y\rangle$. The four laws are those of Involutions of the Endomorphism Ring.

Proposition (self-adjoint and skew-adjoint). The self-adjoint matrices are those with $\sigma(X)^{\mathrm t}\Phi = \Phi X$, equivalently $X^{*} = X$; the skew-adjoint ones are those with $X^{*} = -X$. With $2$ invertible the ring is the sum of the two, and for $\Phi = I$ they are the matrices with $\sigma(X)^{\mathrm t} = \pm X$, the Hermitian and skew-Hermitian matrices of the coefficient involution.

Proof. $X^{*} = X$ is $\Phi^{-1}\sigma(X)^{\mathrm t}\Phi = X$, i.e. $\sigma(X)^{\mathrm t}\Phi = \Phi X$; the rest is Involutive Rings applied to the adjoint involution.

Proposition (the unitary group). The matrices with

$$ X^{*}X = XX^{*} = I, \qquad \text{equivalently } \sigma(X)^{\mathrm t}\Phi X = \Phi , $$

form a subgroup $U_n(R,\Phi,\sigma)$ of $\mathrm{GL}_n(R)$ and are exactly the matrices preserving the form, $\langle Xx,Xy\rangle = \langle x,y\rangle$ for all $x,y$. For $\sigma = \mathrm{id}$ and $\Phi$ symmetric this is the orthogonal group, for $\Phi^{\mathrm t} = -\Phi$ the symplectic group, and for $R = \mathbb{C}$, $\sigma$ the conjugation and $\Phi = I$ the unitary group.

Proof. $\sigma(X)^{\mathrm t}\Phi X = \Phi$ is $X^{*}X = I$; multiplying by $X$ on the left gives $X X^{*} = I$ by the standard group argument, and the preservation is the unitary-element proposition of Involutions of the Endomorphism Ring.

The Trace Pairing

Proposition. With the trace pairing $\beta(X,Y) = \tau(XY)$ of the ring, the adjoint of the left multiplication is the right multiplication, $L_X^{*} = R_X$, and the trace of the adjoint is the $\sigma$-image of the trace,

$$ \tau(X^{*}) = \sigma(\tau(X)) . $$

Hence the trace pairing is $\sigma$-semilinear in the appropriate sense and its restriction to the self-adjoint matrices is a symmetric bilinear form.

Proof. $\beta(L_XY,Z) = \tau(XYZ) = \tau(YZX) = \beta(Y,R_XZ)$ is The Transpose as an Adjoint; $\tau(X^{*}) = \tau(\Phi^{-1}\sigma(X)^{\mathrm t}\Phi) = \tau(\sigma(X)^{\mathrm t}) = \sigma(\tau(X))$, since the trace is invariant under conjugation by $\Phi$ and additive over the diagonal.

Corollary (the two involutions). The adjoint involution $X\mapsto\sigma(X)^{\mathrm t}$ of $\Phi = I$ and the transpose involution $X\mapsto X^{\mathrm t}$ of Matrix Rings with an Involution agree exactly when $\sigma = \mathrm{id}$; for a nontrivial $\sigma$ the adjoint involution is of the second kind and its self-adjoint matrices are the Hermitian ones.

Proof. Comparison of the two formulas; the kind is read off from the action on the centre $R\cdot I$, where $\sigma(X)^{\mathrm t}$ acts by $\sigma$ on the scalars.

Examples

(a) The real orthogonal case. $R = \mathbb{R}$, $\sigma = \mathrm{id}$, $\Phi = I$: $X^{*} = X^{\mathrm t}$, the self-adjoint matrices are the symmetric ones, and $U_n$ is the orthogonal group.

(b) The complex unitary case. $R = \mathbb{C}$, $\sigma$ the conjugation, $\Phi = I$: $X^{*} = \overline{X}^{\mathrm t}$, the self-adjoint matrices are the Hermitian ones, and $U_n$ is the unitary group. The adjoint involution is of the second kind.

(c) The symplectic case. $\Phi = J$ with $J^{\mathrm t} = -J$, $J^2 = -1$: $X^{*} = -JX^{\mathrm t}J$ and $U_n$ is the symplectic group of Matrix Rings with an Involution.

(d) The indefinite case. $\Phi = \operatorname{diag}(1,\dots,1,-1,\dots,-1)$: the self-adjoint matrices are the Hermitian matrices of a signature, and the unitary group is the corresponding indefinite unitary group; the signature is a Part II invariant and is only named here.

Summary

In the matrix ring with a coefficient involution $\sigma$ and an invertible Gram matrix $\Phi$, the adjoint is $X^{*} = \Phi^{-1}\sigma(X)^{\mathrm t}\Phi$, defined by the sesquilinear form $\langle x,y\rangle = \sigma(x)^{\mathrm t}\Phi y$; for $\Phi = I$ it is $\sigma(X)^{\mathrm t}$. The self-adjoint matrices satisfy $\sigma(X)^{\mathrm t}\Phi = \Phi X$, the skew-adjoint ones the same with the opposite sign, and the unitary group $U_n(R,\Phi,\sigma) = \{X : X^{*}X = XX^{*} = I\}$ is the group of matrices preserving the form; the transpose, the orthogonal, the symplectic and the unitary groups are the cases $\Phi = I$ with $\sigma = \mathrm{id}$, a symmetric $\Phi$, an antisymmetric $\Phi$, and $R = \mathbb{C}$ with the conjugation. Under the trace pairing $\beta(X,Y) = \tau(XY)$ the adjoint of the left multiplication is the right multiplication and $\tau(X^{*}) = \sigma(\tau(X))$, so the trace pairing restricts to a symmetric form on the self-adjoint part.

Summary of Notation

Symbol	Meaning
$\sigma$, $\Phi$	Coefficient involution and invertible Gram matrix
$\langle x,y\rangle = \sigma(x)^{\mathrm t}\Phi y$	Sesquilinear form
$X^{*} = \Phi^{-1}\sigma(X)^{\mathrm t}\Phi$	Adjoint of $X$
$\sigma(X)^{\mathrm t}\Phi = \Phi X$	Self-adjoint matrices
$X^{}X = XX^{} = I$	Unitary group $U_n(R,\Phi,\sigma)$
$\sigma(X)^{\mathrm t}\Phi X = \Phi$	Preservation of the form
$\beta(X,Y) = \tau(XY)$	Trace pairing; $L_X^{*} = R_X$
$\tau(X^{*}) = \sigma(\tau(X))$	Trace of the adjoint
$\Phi = I$ / $J$ / indefinite	Orthogonal / symplectic / indefinite unitary cases