I am studying spectral theory of functional analysis. I understand there are deep connections between the spectrum of operators and the eigenvalues of matrices. But I am unable to solve the following such a simple problem yet.
Let $T$ be in a complex-valued matrix space, i.e., $T \in \mathbb{M}_n(\mathbb{C})$, equipped with the operator norm.
Show that the spectrum of $T$ is the set of its eigenvalues $\lambda_i$.
Use the Jordan decomposition to prove that the sequence $(\|T^n\|^\frac{1}{n})_{n\in\mathbb{N}_{\ge 1}}$ converges towards the spectral radius of $T$, i.e., $\|T^n\|^\frac{1}{n} \to \max{|\lambda_i|}$ as $n \to \infty$.
Show that the sequence $(\|T^n\|^\frac{1}{n})_{n\in\mathbb{N}_{\ge 1}}$ also converges, if $T$ is in a Banach algebra.
I think, if $T \in \mathbb{M}_n(\mathbb{C})$, then $\|T\| := \sup \{\|Tx\|: x \in \mathbb{C}^n, \|x\| \le 1\}$. To show the spectrum of $T$, I know they are those $\lambda_i$ who make $(T - \lambda_i I)$ not invertible, then $(T - \lambda_i I) x_i = 0$ has non-trivial solution $x_i \ne 0$. I know the $x_i$ should be the eigenvectors due to $Tx_i = \lambda_i x_i$, and $\lambda_i$ should be the eigenvalues. However, I am not sure if my statement above could be a proof of Q1.
For Q2 and Q3, I have totally no idea. Could anyone help me to solve the problems, and to understand further the concepts behind spectrum and eigenvalues?