Covariance symmetric/positive-definite

Let us call x the vector of Na and Cl concentrations, x the vector of sample means and S the symmetric, positive-definite covariance matrix, i.e., the 2 x 2 matrix with variances on the diagonal and the covariance between Na and Cl concentrations as off-diagonal terms. The equation of the ellipse to be drawn can be written... 

It will be assumed throughout that H and are symmetric positive-definite tensors. We write the mobility as a contravariant tensor, with raised bead indices, to reflect its function the mobility H may be contracted with a covariant force vector Fv (e.g., the derivative of a potential energy) to produce a resulting contravariant velocity Fy. 

Let us now have a general set of linear equations (constraints) (7.1.1), with the regularity condition (7.1.4), and with the partition of variables (7.1.9) into measured (x R ) and unmeasured (y e R ). We thus also know the set (7.3.4) of vectors x obeying the condition of solvability, see (7.3.2). Let x" be the vector of actually measured values. Suppose we know the covariance matrix F of measurement errors it is an / x / symmetric positive definite matrix. If the matrix is diagonal (errors uncorrelated) thus... 

E((e-Co )(e-eo ) ), covariance matrix of measurement errors, symmetric positive definite, / x 7 if not otherwise specified given a priori by the statistical model of measurement submatrices of F (11.2.1), covariance matrices of measurement errors in the components of s, and mj, , respectively... 

Let w be a random A-dimensional vector with a mean n = E v) and a covariance matrix E. Since E is symmetric, positive-definite, E always exists. The Gaussian (normal) distribution of v is... 

Since E is a positive definite symmetric matrix, its trace can be taken as a measure of the estimate error covariance. 

The covariance matrix is factored using the diagonal matrix A and the eigenvector matrix U as U UT. Since 5 is symmetric and positive-definite, the eigenvalues are positive and the eigenvectors orthogonal. The inverse S 1 of S can be expanded as UA 1UT and the transformation... 

A real, symmetric matrix A is called positive definite if x Ax > 0 for every conforming nonzero real vector x. Extend the result of (a) to show that the covariance matrix E in Eq. (4.C-1) is positive definite if the scalar random variables i ,.... Emu are linearly independent, that is, if there is no nonzero m-vector x such that x Eu vanishes over the sample space of the random vector . 

This is achieved through Cholesky factorization, which is a method to simulate multivariate normal returns, based on the assumption that the covariance matrix is symmetric and positive-definite. It is used to ensure the simulated series have a certain desired correlation. 

It can be shown that all symmetric matrices of the form X X and XX are positive semi-definite [2]. These cross-product matrices include the widely used dispersion matrices which can take the form of a variance-covariance or correlation matrix, among others (see Section 29.7). 

The covariance matrix is positive semi-definite and symmetric. Thus, it can be written in terms of eigenvalues and eigenvectors as... 

The verbal interpretation of (5.5) is that the process is continuous — this is the Lindeberg condition. The function a x, t) is the velocity of conditional expectation ( drift vector ), and bjj(x, t) is the matrix of the velocity of conditional covariance ( diffusion matrix ). The latter is positive semidefinite and symmetric as a result of its definition (5.7). 

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