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In computational mechanics, a tangent stiffness matrix is a matrix that describes the stiffness of a system in response to small changes in configuration. It represents tangent in that the energy of the system can be thought of as a high-dimensional surface with the local slope of a plane tangent to it at the given point defined by the tangent stiffness matrix.
The tangent stiffness matrix appears when solving certain problems. For example, the tangent stiffness matrix is the generalization of slope that can be used with Newton's method.