What is contravariant and covariant vector, tensor in. Lorentz covariance, a related concept, is a property of the underlying spacetime manifold. Higher order tensors are in principle handled similarly, but they may be expressed with mixed coordinate types i. The scalar product is a tensor of rank 1,1, which we will denote i and call the identity tensor. Aug 03, 2006 related threads on what different between covariant metric tensor and contravariant metric tensor units if conversion between covariantcontravariant tensors last post.
Tensorbased covariance matrices for object tracking ieee xplore. A di erent metric will, in general, identify an f 2v with a completely di erent ef 2v. Once we start viewing the covariance matrix as a transformationdriven metric tensor, many things become clearer, but one thing becomes extremely puzzling. The metric tensor is g ij and is most easily understood when represented by a square matrix. In this paper, an analytical expression of the covariance matrix of the eigenvalues and eigenvectors of an nd symmetric tensor is derived based on the principles of linear algebra and differential calculus. Note that the metric tensor may be a function of position in the space. For other uses of covariant or contravariant, see covariance and contravariance disambiguation. A manifold equipped with a positivedefinite metric tensor is known as a riemannian manifold. Pdf various metrics for comparing diffusion tensors have been recently proposed in the literature. Although covariance matrices can be positive semide nite in.
A physical quantity is said to be lorentz covariant if it transforms under a given representation of the lorentz group. Understanding the role of the metric in linking the various forms of tensors1 and, more importantly, in di. In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis in physics, a basis is sometimes thought of as a set of reference axes. Chapter 10 vectors and tensors georgia institute of. When studying tensor calculus the distinction between covariance and contravariance may be obscure and is rarely explained visually.
Einstein relatively easy principle of general covariance. Jun 28, 2012 but i would like to have christofell symbols in terms of the metric to be pluged in this equation. We require, that the metric tensor is symmetric g ij g ji, or we consider only the symmetrized tensor. Pythagoras theorem which, written in terms of a metric tensor gab, is1. First we will explain the distinction between the covariant and contravariant components of vectors, thinking of vectorfields where a vector is defined at a point rather than as a. Introduction to tensor calculus for general relativity.
Covariant derivative and metric tensor physics forums. Introduction to tensor calculus for general relativity mit. General covariance and coordinate transformation in classical. The rules for transformation of tensors of arbitrary rank are a generalization of the rules for vector transformation. This collection of notes attempts to demystify some of the vagaries of tensor analysis, to emphasize the connections between the old way of doing things and the new. Metric tensor and the line element phys 500 southern illinois university september 22, 2016 phys 500 southern illinois university metric tensor and the line element september 22, 2016 1 16. Noneuclidean statistics for covariance matrices, with. Comparing the lefthand matrix with the previous expression for s 2 in terms of the covariant components, we see that. Tensors have their applications to riemannian geometry, mechanics, elasticity, theory of relativity, electromagnetic theory and many other disciplines of science and engineering.
It gives me great pleasure to write the foreword to dr. A metric tensor is called positivedefinite if it assigns a positive value gv, v 0 to every nonzero vector v. Nazrul islams book entitled tensors and their applications. On a manifold, a tensor field will typically have multiple, upper and lower indices, where einstein notation is widely used. Requirements 1 the derivative of a tensor must be a tensor. Or, in indices, a tangent vector is by definition a vector that is a linear combination of the coordinate partials.
On covariance propagation of eigenparameters of symmetric nd. Spatial channel covariance estimation for hybrid architectures based on tensor decompositions sungwoo park, anum ali, nuria gonzalezprelcic, and robert w. The covamnet is designed to exploit both the covariance representation and covariance metric based on the distribution consistency for the fewshot classification tasks. A metric for covariance matrices wolfgang forstner, boudewijn moonen institut fur photogrammetrie, universitat bonn nussallee 15, d53115 bonn, email. Pdf power euclidean metrics for covariance matrices with. Is it possible to obtain a scalar from two vectors or two oneforms. Related threads on what different between covariant metric tensor and contravariant metric tensor units if conversion between covariantcontravariant tensors last post. Covariant derivative 22 mar 2012riemannchristoffel curvature tensor. For every mode unfolding matrix, we use the kl transform to achieve the principal components of the column vectors.
We may play this game in the euclidean space en with its \dot inner product. I know the author as a research scholar who has worked with me for several years. An informal discussion on tensor calculus chris tiee abstract. Covariance and contravariance of vectors 5 then the frame f is related to the frame f by the inverse of the jacobian matrix of the coordinate transition. Within a factor a0, the torsion tensor is the electromagnetic eld tensor. An ordinary matrix is a rank 2 tensor, a vector is a rank 1 tensor, and a scalar is rank 0.
A mathematical derivation of the general relativistic schwarzschild metric an honors thesis presented to the faculty of the departments of physics and mathematics east tennessee state university in partial ful. For example, for a tensor of contravariant rank 2 and covariant rank 1. To clarify this we will translate all wellknow vector and matrix manipulations. Tensor which depends on two directions one normal to the area and other that of the force on it. I have a problem that i dont know how to compute the covariance of two tensor. Tensor sparse coding for region covariances 3 the feature vector z usually consists of color information in some preferred colorspace, usually rgb and information about the rst and higher order spatial derivatives of the image intensity, depending on the application intended. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. The definition of contravariant tensor of rankis a simple generalization.
What different between covariant metric tensor and. Covariant and contravariant vectors alok kumar1 iiser, bhopal iti campus gas rahat building govindpura, bhopal 23 india. Our notation will not distinguish a 2,0 tensor t from a 2,1 tensor t, although a notational distinction could be made by placing marrows and ntildes over the symbol, or by appropriate use of dummy indices wald 1984. Tensors can, in general, have rank greater than 2, and. A metric for covariance matrices university of bonn. That is, the matrix that transforms the vector components must be the inverse of the matrix that transforms the basis vectors. We use this repeatedly in the algebra in sections below. In order to change a contravariant vector into a oneform or viseversa so that we can compute the inner product, we use the metric tensor as a lowering or raising operator. Heath jr abstract spatial channel covariance information can replace full instantaneous channel state information for the analog precoder design in hybrid analogdigital architectures. The connection is chosen so that the covariant derivative of the metric is zero.
I mean, prove that covariant derivative of the metric tensor is zero by using metric tensors for gammas in the equation. An example application is in diffusion tensor imaging where a diffusion tensor is a covariance matrix related to the molecular displacement at a particular voxel in the brain, as described in section 2. Intuition being, that given a vector with dxi vi, this will give the length of the vector in our geometry. In multilinear algebra and tensor analysis, covariance and contravariance describe how the. Lorentz covariance has two distinct, but closely related meanings.
Finally we are in a position to see how covariant and contravariant vectors differ in this metric. One important characteristic is the rank of a tensor, which is the number of indicies needed to specify the tensor. A mathematical derivation of the east tennessee state. In writing this thesis i also stumbled upon several discrepancies in the original article. For instance, a rank2 contravariant tensor has the transformation property. Introduction and examples modeling mean structure modeling covariance structure mean and covariance models for tensorvalued data peter ho statistics, biostatistics and the csss university of washington. Covariance and contravariance of vectors semantic scholar. A change of scale on the reference axes corresponds to a change of units in the problem. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold. Pdf distribution consistency based covariance metric. So in order to keep the thesis selfcontained, appendices have been added to introduce these concepts in a degree which su. When the manifold is equipped with a metric, covariant and contravariant indices become very closely related to one another. Abstract vector is a physical quantity and it does not depend on any coordinate system. Now that we have talked about tensors, we need to figure out how to classify them.
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