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Recursive updating the eigenvalue decomposition of a covariance matrix

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The converse holds trivially: if A can be written as LL* for some invertible L, lower triangular or otherwise, then A is Hermitian and positive definite.

A closely related variant of the classical Cholesky decomposition is the LDL decomposition, Some indefinite matrices for which no Cholesky decomposition exists have an LDL decomposition with negative entries in D.

The Cholesky algorithm, used to calculate the decomposition matrix L, is a modified version of Gaussian elimination.

The recursive algorithm starts with i := 1 and Either pattern of access allows the entire computation to be performed in-place if desired.

Loss of the positive-definite condition through round-off error is avoided if rather than updating an approximation to the inverse of the Hessian, one updates the Cholesky decomposition of an approximation of the Hessian matrix itself.

recursive updating the eigenvalue decomposition of a covariance matrix-74recursive updating the eigenvalue decomposition of a covariance matrix-62

The columns of L can be added and subtracted from the mean x to form a set of 2N vectors called sigma points.

The correlation matrix is decomposed, to give the lower-triangular L.

Applying this to a vector of uncorrelated samples u produces a sample vector Lu with the covariance properties of the system being modeled..

For linear systems that can be put into symmetric form, the Cholesky decomposition (or its LDL variant) is the method of choice, for superior efficiency and numerical stability.

Compared to the LU decomposition, it is roughly twice as efficient.