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  "body": "\n## Summary\n\nA tridiagonal matrix is a square matrix whose nonzero entries lie only on the main diagonal and on the two diagonals next to it.\nIt is a [[band matrix|Z58FRZCF8P]] of bandwidth one on each side, and it is both upper and lower [[Hessenberg|YCEEWJTX56]].\nLinear systems with a tridiagonal matrix can be solved with a number of operations proportional to the size, not to its cube.\nSuch matrices arise from finite difference schemes, from three-term recurrences and from the reduction of symmetric matrices for eigenvalue computations.\n\n## Explanations\n\n### L1 human\n\nPicture a square matrix where only a narrow strip of three diagonals is filled and everything else is zero.\nFor example, $\\begin{pmatrix} 2 \u0026 -1 \u0026 0 \\\\ -1 \u0026 2 \u0026 -1 \\\\ 0 \u0026 -1 \u0026 2 \\end{pmatrix}$ is tridiagonal.\nThis one comes from approximating a second derivative at evenly spaced points: each point only talks to its two neighbours.\nBecause each unknown meets only its neighbours, elimination proceeds row by row without creating new nonzero entries.\n",
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