# Cholesky decomposition and rotation matrix inverse

I implemented three methods for inversion of a matrix, all are classic. I wanted to test for the most generalized method, while taking efficiency into account.

For Cholesky decomposition, which is relatively efficient, the matrix $A$ should be positive or semi positive definite. It is known for a matrix $A$ to be positive definite, its eigenvalues must be positive. Now, in the case of rotation matrices, their eigenvalues represent modes "think of frequency, if that is a proper interpretation", and therefore are complex "or real". In such case...is there a such thing as a reliable Cholesky decomposition for the inversion of an arbitrary rotation matrix.

Note: $R^{-1}=R^T$ for rotation matrices "orthogonal", but still, i would like to test for explicit inversion using Cholesky method with forward and backward substitution

• Cholesky works on symmetric positive definite matrices. An arbitrary rotation matrix is not necessarily symmetric. (For example, a Givens rotation.) Also, what is your measurement for efficiency? Operation count? – artificial_moonlet Jun 3 '15 at 17:21
• The measure for efficiency is rather an optimization effort, meaning anything less than what Cholesky $< O(n^3)$ .So based on your answer , just go head and use LU as a generic method for an inversion of a matrix. Is there any more efficient methods..say in the order of $O(n^2)$ – Sam Gomari Jun 4 '15 at 4:04
• Operation count is measured in the approximate number of flops-- e.g., $O(n), O(n^2)$, etc. As for a general inversion method of $O(n^2)$-- well, that's the million dollar question in numerical linear algebra. – artificial_moonlet Jun 4 '15 at 15:04