📐 Jordan Canonical Form Theorem

Every linear operator on a finite-dimensional complex vector space has a unique Jordan canonical form (up to ordering of blocks).

Proof: **Existence:** By the primary decomposition theorem, $V = V_1 \\oplus \\cdots \\oplus V_k$ where $V_i$ is the generalized eigenspace for $\\lambda_i$.

On each $V_i$, $(T - \\lambda_i I)$ is nilpotent. For nilpotent operators, there exists a Jordan basis giving blocks of the form $J_m(0)$.

Combi...

From: Advanced Linear Algebra

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