Let
be a
random vector. The **covariance
matrix** of
,
or variance-covariance matrix of
,
is denoted by
.
It is defined as
follows:provided
the above expected values exist and are
well-defined.

It is a multivariate generalization of the definition of variance for a scalar random variable :

Let , ..., denote the components of the vector . From the definition of , it can easily be seen that is a matrix with the following structure:

Therefore, the covariance matrix of is a square matrix whose generic -th entry is equal to the covariance between and .

Since when , the diagonal entries of the covariance matrix are equal to the variances of the individual components of .

Example Suppose is a random vector with components and . LetBy the symmetry of covariance, it must also be Therefore, the covariance matrix of is

The covariance matrix of a random vector can be computed as follows:

Proof

The above formula can be derived as follows:

This formula also makes clear that the covariance matrix exists and is well-defined only as long as the vector of expected values and the matrix of second cross-moments exist and are well-defined.

The following subsections contain more details about the covariance matrix.

Let be a constant vector and let be a random vector. Then,

Proof

This is a consequence of the fact that (by linearity of the expected value):

Let be a constant matrix and let be a random vector. Then,

Proof

This is easily proved using the fact that (by linearity of the expected value):

Let be a constant vector, be a constant matrix and a random vector. Then, combining the two properties above, one obtains

The covariance matrix is a symmetric matrix, that is, it is equal to its transpose:

The covariance matrix is a positive-semidefinite matrix, that is, for any vector :This is easily proved using the Multiplication by constant matrices property above:where the last inequality follows from the fact that variance is always positive.

Let and be two constant vectors and a random vector. Then, the covariance between the two linear transformations and can be expressed as a function of the covariance matrix:

Proof

This can be proved as follows:

The term covariance matrix is sometimes also used to refer to the matrix of covariances between the elements of two vectors.

Let
be a
random vector and
be a
random vector. The **covariance matrix** between
and
,
or cross-covariance between
and
is denoted by
.
It is defined as
follows:provided
the above expected values exist and are well-defined.

It is a multivariate generalization of the definition of covariance between two scalar random variables.

Let , ..., denote the components of the vector and , ..., denote the components of the vector . From the definition of , it can easily be seen that is a matrix with the following structure:Note that is not the same as . In fact, is a matrix equal to the transpose of :

Below you can find some exercises with explained solutions.

Let be a random vector and denote its components by and . The covariance matrix of isCompute the variance of the random variable defined as

Solution

By using a matrix notation, can be written aswhere we have definedTherefore, the variance of can be computed using the formula for the covariance matrix of a linear transformation:

Let be a random vector and denote its components by , and . The covariance matrix of isCompute the following covariance:

Solution

Using the bilinearity of the covariance operator, we obtainThe same result can be obtained using the formula for the covariance between two linear transformations. Definingwe have

Let be a random vector whose covariance matrix is equal to the identity matrix:Define a new random vector as follows:where is a matrix of constants such thatDerive the covariance matrix of .

Solution

By the formula for the covariance matrix of a linear transformation, we have

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