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  "body": "\n## Definition\n\nLet $F$ be a field, let $V$ be a vector space over $F$, and let $T \\colon V \\to V$ be an [[endomorphism|0J7H0YBFDJ]].\nA scalar $\\lambda \\in F$ is an *eigenvalue* of $T$ when there is a vector $v \\in V$ with $v \\ne 0$ and\n$$T(v) = \\lambda v.$$\nAn eigenvalue of an $n \\times n$ matrix $A$ over $F$ is an eigenvalue of the map $F^n \\to F^n$, $x \\mapsto Ax$.\n\n## Remarks\n\nSo $\\lambda$ is an eigenvalue of $T$ if and only if $T - \\lambda \\id_V$ is not injective.\nIn finite dimensions, this holds if and only if $\\det(T - \\lambda \\id_V) = 0$.\nThe eigenvalues of the diagonal matrix $\\begin{pmatrix} 2 \u0026 0 \\\\ 0 \u0026 3 \\end{pmatrix}$ are $2$ and $3$.\nThe rotation of $\\R^2$ by $90$ degrees has no real eigenvalue.\n\n## Explanations\n\n### L1 human\n\nMost vectors change direction when a linear map acts on them.\nAn eigenvalue is a scaling factor $\\lambda$ such that some nonzero vector only gets stretched by $\\lambda$, without a change of its line.\nThese special directions often reveal the structure of the map.\n",
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