eigen-decomposition

Eigen-decomposition is a means of factoring one square matrix into three matrices that produce the original matrix when multiplied. The term is also used for the result, the three matrices, which together have properties that are useful for some kinds of analysis and can also be used to save some kinds of calculation effort. Not all square matrices can undergo eigen-decomposition.

Eigen-decomposition begins by determining the linearly-independent eigenvectors of a square matrix and constructing a square matrix of them, which is termed an eigen-basis. If there is such an eigen-basis, it is invertible, and the eigen-decomposition is possible, and these two matrices (the eigen-basis and its inverse) are two of the three matrices in the decomposition. The other matrix of the decomposition is the product of three matrices: the inverse of the eigen-basis times the original matrix times the eigen-basis. This last member of the decomposition is always a diagonal matrix and building it is termed diagonalization. The original matrix is equal to the eigen-basis times the diagonal matrix times the inverse of the eigen-basis.

Eigen-decomposition is used in numerous of fields of science including both quantum mechanics and the classical dynamics of oscillating contraptions.