Properties
For and, where X is a random p-dimensional variable and Y a random q-dimensional variable, the following basic properties apply:
- is positive-semidefinite and symmetric.
- If p = q, then
- If and are independent or uncorrelate, then
where and are random p×1 vectors, is a random q×1 vector, is a q×1 vector, is a p×1 vector, and and are q×p matrices.
This covariance matrix is a useful tool in many different areas. From it a transformation matrix can be derived, called a whitening transformation, that allows one to completely decorrelate the data or, from a different point of view, to find an optimal basis for representing the data in a compact way (see Rayleigh quotient for a formal proof and additional properties of covariance matrices). This is called principal components analysis (PCA) and the Karhunen-Loève transform (KL-transform).
Read more about this topic: Covariance Matrix
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)