December 2015 Archives

Last time we took a first look at the Cholesky decomposition, being the unique way to represent a positive definite symmetric square matrix M as the product of a lower triangular matrix L having strictly positive elements on its leading diagonal, and by definition zeroes everywhere above it, with its upper triangular transpose L^T, formed by switching the row and column indices of each element so that L^T_ij = L_ji

    M = L × L^T

Finally we noted that, like the Jacobi decomposition, the Cholesky decomposition is useful for solving simultaneous equations, inverting matrices and so on, but left the implementation details to this post.

Some time ago, we took a look at the Jacobi decomposition which is used to find the eigensystems of real symmetric matrices being, for a matrix M, those vectors v_i and numbers λ_i that satisfy

    M × v_i = λ_i × v_i

This representation of a matrix turned out to have some tremendously useful consequences, such as dramatically simplifying both the solution of simultaneous linear equations and the calculation of functions of matrices like the inverse and, crucially in the context of this post, the square root.