30.1 The Matrix Algebra
Definition 30.1.1 (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 30.1.2.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 30.1.3.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 30.1.2, $\sigma(x)$ is finite. By Proposition 29.3.2, there exists $y \in M_{n}(\complex)$ such that $x = \exp(y)$. In which case, $x \in G_{0}(M_{n}(\complex))$.$\square$
Proposition 30.1.4.label Let $n \in \natp$, then
- (1)
$M_{n}(\complex)$ admits no nontrivial two-sided ideals.
- (2)
$M_{n}(\complex)$ admits no multiplicative functionals.
Proof. Let $x = (x_{ij}) \in M_{n}(\complex) \setminus \bracs{0}$, then there exists $1 \le i, j \le n$ such that $x_{ij}\ne 0$. In which case, for any $1 \le k, l \le n$ and $\lambda \in \complex$, there exists $y_{k}, z_{l} \in M_{n}(\complex)$ such that $y_{k}xz_{l}$ is the matrix with $\lambda$ on its $(k, l)$ entry and $0$ everywhere else. Therefore every non-trivial two-sided ideal of $M_{n}(\complex)$ is $M_{n}(\complex)$.$\square$
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