The covariance matrix of a random vector is a square matrix that contains all the covariances between the entries of the vector.
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Here is a definition.
It is a multivariate generalization of the definition of variance for a scalar random variable :
Let , ..., denote the components of the vector .
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 entries of .
Here is an example.
Suppose that is a random vector with components and .
By the symmetry of covariance, it must also be
Therefore, the covariance matrix of is
The covariance matrix of a random vector can be computed using the formula
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,
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,
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, by combining the two properties above, we obtain
The covariance matrix is a symmetric matrix, that is, it is equal to its transpose:
The proof is as follows:
The covariance matrix is a positive-semidefinite matrix, that is, for any vector :
This is easily proved by 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:
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 is
Compute the variance of the random variable defined as
By using a matrix notation, can be written aswhere we have definedTherefore, the variance of can be computed by 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 is
Compute the following covariance:
Using the bilinearity of the covariance operator, we obtainThe same result can be obtained by 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 that
Derive the covariance matrix of .
By the formula for the covariance matrix of a linear transformation, we have
Please cite as:
Taboga, Marco (2021). "Covariance matrix", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/fundamentals-of-probability/covariance-matrix.
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