35.1 The Matrix Algebra
Definition 35.1.1 (Matrix Algebra).label Let $n \in \natp$ and $M_{n}(\complex)$ be the set of all $n \times n$ matrices with entries in $\complex$, then $M_{n}(\complex)$ equipped with the operator norm is the matrix algebra over $\complex$.
Proposition 35.1.2.label Let $n \in \natp$ and $x \in M_{n}(\complex)$, then $\sigma(x)$ is the set of eigenvalues of $x$.
Proof. By the rank-nullity theorem, $\lambda - x$ is invertible if and only if $\lambda$ is not an eigenvalue of $x$.$\square$
Proposition 35.1.3.label Let $n \in \natp$, then
- (1)
For each $x \in G(M_{n}(\complex))$, there exists $y \in M_{n}(\complex)$ such that $x = \exp(y)$.
- (2)
$G(M_{n}(\complex)) = G_{0}(M_{n}(\complex))$.
- (3)
$I(M_{n}(\complex))$ is trivial.
Proof. (1), (2): By Proposition 35.1.2, $\sigma(x)$ is finite. By Proposition 33.3.2, there exists $y \in M_{n}(\complex)$ such that $x = \exp(y)$. In which case, $x \in G_{0}(M_{n}(\complex))$.$\square$
Proposition 35.1.4.label Let $n \in \natp$. For each $y \in M_{n}(\complex)$, let
then the following are equivalent:
- (1)
$\phi_{y} \in S(M_{n}(\complex))$.
- (2)
$y \ge 0$ and $\text{tr}(y) = 1$.
Proposition 35.1.5.label Let $n \in \natp$. For each $y \in M_{n}(\complex)$, let
then the following are equivalent:
- (1)
$\phi_{y}$ is a pure state of $M_{n}(\complex)$.
- (2)
$y$ is a projection operator with rank $1$.
- (3)
There exists $v \in \complex^{n}$ with $\norm{v}_{\complex^n}= 1$ such that $\dpn{x, \phi_y}{M_n(\complex)}= \dpn{xv, v}{\complex^n}$ for all $x \in M_{n}(\complex)$.
Proof. (1) $\Leftrightarrow$ (2): By Proposition 35.1.4, $y \ge 0$ with $\text{tr}(y) = 1$. Via an orthogonal change of coordinates, assume without loss of generality that $y$ is diagonal. In which case, $y$ corresponds to an extreme point of $S(M_{n}(\complex))$ if and only if it is of rank $1$. As $\text{tr}(y) = 1$, $y$ is a projection.
(2) $\Rightarrow$ (3): Let $v \in \complex^{n}$ be a unit eigenvector of $y$, then for each $x \in M_{n}(\complex)$,
$\square$
then $A$ is a self-adjoint element of $M_{2}(\complex)$. By Proposition 35.1.5, the mapping $T \mapsto \dpn{Tv, v}{\complex^2}$ is a pure state on $M_{2}(\complex)$ for every unit vector $v \in \complex^{2}$. In particular, if $v = (\sqrt{2}, \sqrt{2})/2$, then $\dpn{Tv, v}{\complex^2}= 0 \not\in \sigma_{M_2(\complex)}(A)$. Therefore
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