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  "body": "\n## Definition\n\nLet $F$ be a field and let $A$ be an $n \\times n$ matrix over $F$.\nThe matrix $A$ is *diagonalisable* over $F$ when there is an invertible $n \\times n$ matrix $P$ over $F$ such that $P^{-1} A P$ is a [[diagonal matrix|044JNC7ZAA]].\nAn endomorphism of a finite-dimensional space is diagonalisable when its matrix in some basis is diagonal.\n\n## Remarks\n\nSo $A$ is diagonalisable if and only if it is [[similar|0E89HQATM8]] to a diagonal matrix.\nThe diagonal entries of $P^{-1} A P$ are the eigenvalues of $A$, and the columns of $P$ are eigenvectors.\nThe matrix $\\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix}$ is not diagonalisable over any field.\nOver $\\R$, the rotation matrix $\\begin{pmatrix} 0 \u0026 -1 \\\\ 1 \u0026 0 \\end{pmatrix}$ is not diagonalisable; over $\\C$, it is.\n\n## Explanations\n\n### L1 human\n\nA diagonalisable matrix becomes a diagonal matrix in a suitable basis.\nIn that basis, the map just stretches each coordinate axis by a factor.\nThis makes powers, exponentials and other functions of the matrix easy to compute.\n",
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