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  "body": "\n## Summary\n\nThe stiffness matrix is the matrix of the linear system that the [[finite element method|98GZ9DVHTA]] produces from the bilinear form of a boundary value problem.\nIts entries are the values of the form on pairs of basis functions.\nBecause finite element basis functions have small supports, the matrix is [[sparse|JGACZYEEY7]].\nFor symmetric coercive problems such as the Poisson equation it is [[symmetric|0VR671PHE5]] and [[positive definite|0KX66CZMBB]].\nThe name comes from structural mechanics, where the matrix links the displacements of the nodes with the forces on them.\nSee the [[definition|DPR9DJRPAF]].\n\n## Explanations\n\n### L1 human\n\nThink of a chain of springs whose ends are tied to walls.\nIf you push one joint, the forces on that joint and on its two neighbours change, and nothing else.\nThe stiffness matrix records these numbers: row $i$ says how the force at joint $i$ depends on the positions of all joints.\nFor equal springs in a chain, each row has $2$ on the diagonal and $-1$ next to it, and zeros elsewhere.\n",
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