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Orbital position vector, orbital velocity vector, other orbital elements. In astrodynamics and celestial dynamics, the orbital state vectors (sometimes state vectors) of an orbit are Cartesian vectors of position and velocity that together with their time () uniquely determine the trajectory of the orbiting body in space.
Throughout this article, boldfaced unsubscripted and are used to refer to random vectors, and Roman subscripted and are used to refer to scalar random variables.. If the entries in the column vector = (,, …,) are random variables, each with finite variance and expected value, then the covariance matrix is the matrix whose (,) entry is the covariance [1]: 177 ...
Formally, a multivariate random variable is a column vector = (, …,) (or its transpose, which is a row vector) whose components are random variables on the probability space (,,), where is the sample space, is the sigma-algebra (the collection of all events), and is the probability measure (a function returning each event's probability).
A vector's components change scale inversely to changes in scale to the reference axes, and consequently a vector is called a contravariant tensor. A vector, which is an example of a contravariant tensor, has components that transform inversely to the transformation of the reference axes, (with example transformations including rotation and ...
The sample covariance matrix (SCM) is an unbiased and efficient estimator of the covariance matrix if the space of covariance matrices is viewed as an extrinsic convex cone in R p×p; however, measured using the intrinsic geometry of positive-definite matrices, the SCM is a biased and inefficient estimator. [1]
In probability theory and statistics, a cross-covariance matrix is a matrix whose element in the i, j position is the covariance between the i-th element of a random vector and j-th element of another random vector. A random vector is a random variable with multiple dimensions. Each element of the vector is a scalar random variable.
The covariance matrix of an random vector is an matrix whose (,) th element is the covariance between the i th and the j th random variables. [ 2 ] : p.372 Unlike in the case of real random variables, the covariance between two random variables involves the complex conjugate of one of the two.
With any number of random variables in excess of 1, the variables can be stacked into a random vector whose i th element is the i th random variable. Then the variances and covariances can be placed in a covariance matrix, in which the (i, j) element is the covariance between the i th random variable and the j th one.