covariance - Properties, Covariance matrices

A term describing mathematical equations whose form is identical in different co-ordinate systems; also called form invariance. It is an essential property of the equations of theories of gravitation and nuclear forces.

In probability theory and statistics, the covariance between two real-valued random variables X and Y, with expected values E(X) = μ and E(Y) = ν is defined as:

where E is the expected value. This can also be written:

Intuitively, covariance is the measure of how much two variables vary together (as distinct from variance, which measures how much a single variable varies). If two variables tend to vary together (that is, when one of them is above its expected value, then the other variable tends to be above its expected value too), then the covariance between the two variables will be positive.

On the other hand, if when one of them is above its expected value, the other variable tends to be below its expected value, then the covariance between the two variables will be negative.

Properties

If X, Y are real-valued random variables and a, b are constant ("constant" in this context means non-random), then the following facts are a consequence of the definition of covariance:

For sequences X1, ..., Xn and Y1, ..., Ym of random variables, we have

For a sequence X1, ..., Xn of random variables, we have

Covariance matrices

For column-vector valued random variables X and Y with respective expected values μ and ν, and m and n scalar components respectively, the covariance is defined to be the m×n matrix

For vector-valued random variables, cov(X, Y) and cov(Y, X) are each other's transposes.