Relation To Polar Decomposition
Consider with the Cartan involution . Then is the real Lie algebra of skew-symmetric matrices, so that, while is the subspace of symmetric matrices. Thus the exponential map is a diffeomorphism from onto the space of positive definite matrices. Up to this exponential map, the global Cartan decomposition is the polar decomposition of a matrix. Notice that the polar decomposition of an invertible matrix is unique.
Read more about this topic: Cartan Decomposition
Famous quotes containing the words relation to, relation and/or polar:
“Unaware of the absurdity of it, we introduce our own petty household rules into the economy of the universe for which the life of generations, peoples, of entire planets, has no importance in relation to the general development.”
—Alexander Herzen (1812–1870)
“When needs and means become abstract in quality, abstraction is also a character of the reciprocal relation of individuals to one another. This abstract character, universality, is the character of being recognized and is the moment which makes concrete, i.e. social, the isolated and abstract needs and their ways and means of satisfaction.”
—Georg Wilhelm Friedrich Hegel (1770–1831)
“In time your relatives will come to accept the idea that a career is as important to you as your family. Of course, in time the polar ice cap will melt.”
—Barbara Dale (b. 1940)