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  "body": "\n## Statement\n\nLet $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \\colon V \\to V$ be a linear map.\nAssume that the characteristic polynomial of $T$ splits over $F$ as $p_T(t) = (t - \\lambda_1)^{m_1} \\cdots (t - \\lambda_k)^{m_k}$ with pairwise distinct $\\lambda_i$.\nThen\n$$V = V_{\\lambda_1} \\oplus \\cdots \\oplus V_{\\lambda_k},$$\nwhere $V_{\\lambda_i}$ is the [[generalised eigenspace|0AJ035ZTZE]] of $\\lambda_i$, and $\\dim V_{\\lambda_i} = m_i$.\n\n## Remarks\n\nThe hypothesis holds for every $T$ when $F$ is algebraically closed, for example $F = \\C$.\nEach $V_{\\lambda_i}$ is $T$-invariant, and $T - \\lambda_i \\id$ is nilpotent on it.\nA proof uses the coprime polynomials $(t - \\lambda_i)^{m_i}$ and the [[Cayley–Hamilton theorem|0F765KFC2J]].\nThis [[direct sum|0XYYT5KNTX]] decomposition is the first step towards the [[Jordan normal form|085HQ5G5TE]].\n\n## Explanations\n\n### L1 human\n\nOver $\\C$, the space splits into pieces, one for each eigenvalue.\nOn each piece, the map is $\\lambda$ times the identity plus a nilpotent part.\nSo a hard problem breaks into simpler problems, one for each eigenvalue.\n",
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