December 2015 Archives

You've Got Class, Mr Cholesky!

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 LT, formed by switching the row and column indices of each element so that LTij = Lji

    M = L × LT

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.

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Are You Definitely Positive, Mr Cholesky?

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 vi and numbers λi that satisfy

    M × vi = λi × vi

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.

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