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
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