Euclidean Jordan Algebra (3) - Linear Transformations

For any element $a \in V$, the Lyapunov transformation $L_a : V \rightarrow V$ and the quadratic representation $P_a : V \rightarrow V$ are defined as

$$ \begin{aligned} L_a(x) & := a \circ x, \\ P_a(x) & := (2L_a^2 - L_{a^2})(x) = 2a \circ (a \circ x) - a^2 \circ x, \end{aligned} $$ for all $x \in V$. These transformations are linear and self-adjoint on $V$. In the following example, we describe these transformations in the Euclidean Jordan algebras defined in the above.

Example 4. 

1. For $\mathbb{R}^n$, the above transformations are given by $L_a(x) = a \ast x$ and $P_a(x) = a^2 \ast x$. Here, note that $x \ast y$, a component-wise product, can be regarded as $ \operatorname{Diag}(x)y$ where $\operatorname{Diag}(x)$ denotes a diagonal matrix of size $n$ whose diagonal entries are the entries of $x$.
2. For $\mathcal{S}^n$, the above transformations are given by $L_A(X) = \frac{1}{2}(AX + XA)$ and $P_A(X) = AXA$.
3. For $\mathcal{L}^n$, the above transformations are
   $$ L_a(x) = \begin{pmatrix} a_1 & \bar{a}^{\top} \\ \bar{a} & a_1 I \end{pmatrix} \begin{pmatrix} x_1 \\ \bar{x} \end{pmatrix}, \qquad P_a(x) = \begin{pmatrix} \Vert a \Vert^2 & 2a_1 \bar{a}^{\top} \\ 2a_1 \bar{a} & (a_1^2 - \Vert \bar{a} \Vert^2)I + 2\bar{a}\bar{a}^{\top} \end{pmatrix} \begin{pmatrix} x_1 \\ \bar{x} \end{pmatrix}. $$

In any Euclidean Jordan algebra $V$, one con define automorphism groups in the following way:

Let $\Lambda : V \rightarrow V$ be a linear transformation. Then, $\Lambda$ is an algebra automorphism if it is invertible and $\Lambda(x \circ y) = \Lambda(x) \circ \Lambda(y)$ for all $x,\, y \in V$. The set of all algebra automorphisms of $V$ is denoted by $\operatorname{Aut}(V)$. 

A linear transformation $\Gamma : V \rightarrow V$ is a (symmetric) cone automorphism if $\Gamma(V_+) = V_+$. Such automorphism is automatically invertible. The set of all cone automorphisms of $V$ is denoted by $\operatorname{Aut}(V_+)$. Is is immediate that $\operatorname{Aut}(V) \subseteq \operatorname{Aut}(V_+)$. Moreover, if $\Gamma \in \operatorname{Aut}(V_+)$, then so are $\Lambda^{-1}$ and $\Lambda^{\top}$.

Example 5. 

1. For $\mathbb{R}^n$, it is easily seen that $\operatorname{Aut}(\mathbb{R}^n)$ consists of permutation matrices, and any element in $\operatorname{Aut}(\mathbb{R}_+^n)$ has a form $DP$, where $P$ is a permutation matrix and $D$ is a diagonal matrix with positive diagonal entries.
2. For $\mathcal{S}^n$, it is known that corresponding to any $\Lambda \in \operatorname{Aut}(\mathcal{S}^n)$, there exists an orthogonal matrix $U \in \mathbb{R}^{n \times n}$ such that $\Lambda(X) = UXU^{\top}$ for all $X \in \mathcal{S}^n$. Also, for $\Gamma \in \operatorname{Aut}(\mathcal{S}_+^n)$, there exists an invertible matrix $Q \in \mathbb{R}^{n \times n}$ such that $\Gamma(X) = QXQ^{\top}$ for all $X \in \mathcal{S}^n$.
3. For $\mathcal{L}^n$, it is known that any algebra automorphism $\Lambda$ can be written as $\Lambda = \left( \begin{smallmatrix} 1 & 0 \\ 0 & U \end{smallmatrix} \right)$, where $U$ is an $(n-1) \times (n-1)$ orthogonal matrix. The explicit description of $\operatorname{Aut}(\mathcal{L}_+^n)$ is not known, yet. However, $\Gamma \in \operatorname{Aut}(\mathcal{L}_+^n)$ if and only if there exists $\mu > 0$ such that $\Gamma^{\top}J\Gamma = \mu J$, where $J = \operatorname{diag}(1, -1, \ldots, -1) \in \mathbb{R}^{n \times n}$. 

A linear transformation $\Phi : V \rightarrow V$ is said to be orthogonal if $\left< \Phi(x),\, \Phi(y) \right> = \left< x,\, y \right>$ for all $x, y \in V$. The set of all orthogonal linear transformations is denoted by $\operatorname{Orth}(V)$.

A linear transformation $\Phi : V \rightarrow V$ is is doubly stochastic if $\Phi$ is positive (i.e., $\Phi(V_+) \subseteq V_+$), unital (i.e., $\Phi(e) = e$), and trace preserving (i.e., $\operatorname{tr}(\Phi(x)) = \Phi(x)$ for all $x \in V$). We denote the set of all doubly stochastic linear transformations by $\operatorname{DS}(V)$.

For a Euclidean Jordan algebra $V$, the followings hold:

• The positivity of $\Phi : V \rightarrow V$ is equivalent to that of $\Phi^{\top} : V \rightarrow V$.
• When $V$ carries a canonical inner product, the trace preserving (unital) property of $\Phi$ is equivalent to the unital  (trace preserving) property of $\Phi^{\top}$. In particular, $\Phi$ is doubly stochastic if and only if its transpose is doubly stochastic.
• When $V$ carries a canonical inner product, it is known that $\operatorname{Aut}(V) = \operatorname{Aut}(V_+) \cap \operatorname{Orth}(V)$. Furthermore, if $V$ is simple, we have $\operatorname{Aut}(V) = \operatorname{Aut}(V_+) \cap \operatorname{DS}(V)$.