In a variance-covariance matrix, what do the diagonal elements represent?

In a variance-covariance matrix, what do the diagonal elements represent?

• A. The covariances between pairs of variables.
• B. The means of the variables.
• C. The variances of each variable.
• D. The standard deviations of each variable.

Answer: (C)

The diagonal elements are the variances of each variable because Cov(X_i, X_i) equals Var(X_i). A variance-covariance matrix is built from Cov(X_i, X_j) for every pair of variables, so when the two variables are the same (i = j) you get the variance. The off-diagonal elements are the covariances between different variables. The standard deviation is the square root of the variance, so it does not appear as a diagonal entry. Means aren’t part of this matrix either. For two variables, you’d see Var(X) and Var(Y) on the diagonal and Cov(X, Y) in the off-diagonal positions. Therefore, the diagonal entries represent the variances of each variable.

The diagonal elements are the variances of each variable because Cov(X_i, X_i) equals Var(X_i). A variance-covariance matrix is built from Cov(X_i, X_j) for every pair of variables, so when the two variables are the same (i = j) you get the variance. The off-diagonal elements are the covariances between different variables. The standard deviation is the square root of the variance, so it does not appear as a diagonal entry. Means aren’t part of this matrix either. For two variables, you’d see Var(X) and Var(Y) on the diagonal and Cov(X, Y) in the off-diagonal positions. Therefore, the diagonal entries represent the variances of each variable.