Chapter 7.5, Problem 13E

The sample covariance matrix is a generalization of a formula for the variance of a sample of N scalar measurements, say, t₁,…,t_N. If m is the average of t₁,…,t_N, then the sample variance is given by

$\frac{1}{N-1}{\displaystyle \underset{k=1}{\overset{n}{\sum }}{\left(t_k-m\right)}^2}$    (1)

Show how the sample covariance matrix, S, defined prior to Example 3, may be written in a form similar to (1). [Hint: Use partitioned matrix multiplication to write S as 1/(N − 1) times the sum of N matrices of size p × p. For 1 ≤ k ≤ N, write X_k − M in place of ${\hat{\text{X}}}_k$.]