![]() A more general downdating problem is that we are givenĪnd the Cholesky factorization and wish to obtain the Cholesky factor of. An example is the problem of downdating the Cholesky factorization, where in the simplest case we have the Cholesky factorization of a symmetric positive definite and want the Cholesky factorization of, which is assumed to be symmetric positive definite. Pseudo-orthogonal matrices arise in hyperbolic problems, that is, problems where there is an underlying indefinite scalar product or weight matrix. Such a matrix must satisfy, which means that, and any such orthogonal matrix is pseudo-orthogonal. This form of allows us to conveniently characterize matrices that are both orthogonal and pseudo-orthogonal. We assume that has this form throughout the rest of this article. īy permuting rows and columns in (1) we can arrange that It follows that if is an eigenvalue of then is also an eigenvalue and it has the same algebraic and geometric multiplicities as. Įquation (2) shows that is similar to the inverse of its transpose and hence (since every matrix is similar to its transpose) similar to its inverse. The Lorentz group, representing symmetries of the spacetime of special relativity, corresponds to matrices with. What are some examples of pseudo-orthogonal matrices? For and, is of the form Since is orthogonal, this equation implies that and hence that ![]() Furthermore, is clearly nonsingular and it satisfies It is easy to show that is also pseudo-orthogonal. ![]() A matrix satisfying (1) is also known as a -orthogonal matrix, where is another notation for a signature matrix. ![]()
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