# Precision matrix

The precision matrix of a random vector is the inverse of its covariance matrix.

## Synonyms

The precision matrix is sometimes called concentration matrix.

## Definition

The following is a precise definition.

Definition Let be a random vector. Let be its covariance matrix:If is invertible, then the precision matrix of is the matrix defined as

When is a random variable (), then the precision matrix becomes a scalar and it is equal to the reciprocal of the variance of . In this case, it is often denoted by the lowercase letter :and it is simply called the precision of .

Thus, in the univariate case precision is inversely proportional to variance: when variance tends to infinity, we have zero precision; on the contrary, when variance tends to zero, we have infinite precision.

## Precision matrix and normal densities

The joint probability density function of a multivariate normal random vector is often written in terms of its precision matrix.

If has a multivariate normal distribution with mean and covariance matrix , then its joint probability density function is

By using the precision matrix, this can be written asbecause, by elementary properties of the determinant, we have that

In the univariate case, when is a normal random variable with mean and variance , the densitybecomes

Parametrizing a normal density in terms of its precision matrix often has significant advantages. For example, it can simplify the algebra of calculations involving normal densities. Or, when the values of a multivariate normal density need to be computed several times by numerical methods, employing the precision matrix can spare the computationally burdensome task of performing several matrix inversions to calculate .

## More details

You can read more details about covariance matrices in the lecture entitled Covariance matrix.