Stiffness matrix

For the stiffness tensor in solid mechanics, see Hooke's law#Matrix representation (stiffness tensor).

In the finite element method and in analysis of spring systems, a stiffness matrix, K, is a symmetric positive-semidefinite matrix that generalizes the stiffness of Hooke's law to a matrix, describing the stiffness of between all of the degrees of freedom so that

$\mathbf {F} =-K\mathbf {x}$

where F and x are the force and the displacement vectors, and

$U={\frac {1}{2}}\;\mathbf {x} ^{\top }\!K\mathbf {x}$

is the system's total potential energy.

For a simple spring network, the stiffness matrix is a Laplacian matrix (in order to enforce Newton's third law) describing the connectivity graph between degrees of freedom. Off-diagonal entries contain $k_{ij}$, the negative stiffness of the spring connecting degree-of-freedom i to j. For example,

$K=\left({\begin{array}{rrrrrr}13&-1&0&0&-12&0\\-1&3&-1&0&-1&0\\0&-1&2&-1&0&0\\0&0&-1&3&-1&-1\\-12&-1&0&-1&14&0\\0&0&0&-1&0&1\\\end{array}}\right)$

See also

• Mass matrix
• Hooke's law