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  "body": "\n## Definition\n\nLet $F$ be a field, let $V$ be a vector space over $F$, let $T \\colon V \\to V$ be an endomorphism, and let $\\lambda \\in F$.\nAn *eigenvector* of $T$ for $\\lambda$ is a vector $v \\in V$ with $v \\ne 0$ and $T(v) = \\lambda v$.\nThen $\\lambda$ is an [[eigenvalue|0BQAA2BK3Q]] of $T$.\n\n## Remarks\n\nEach nonzero multiple of an eigenvector is again an eigenvector for the same eigenvalue.\nFor $A = \\begin{pmatrix} 2 \u0026 1 \\\\ 0 \u0026 3 \\end{pmatrix}$, the vector $(1, 1)^T$ is an eigenvector for $3$, because $A (1, 1)^T = (3, 3)^T$.\nThe zero vector is never an eigenvector, by convention.\n\n## Explanations\n\n### L1 human\n\nAn eigenvector keeps its line under the map: the map only stretches it, shrinks it or flips it.\nThe stretch factor is the eigenvalue.\nConsider a reflection of the plane in a line.\nThe vectors on the line are eigenvectors for $1$, and the vectors perpendicular to it are eigenvectors for $-1$.\n",
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