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 t_1...........t_N , then the sample variance is given by

Show how the sample covariance matrix, S, defined prior to Example 3, may be written in a form similar to (1). Use partitioned matrix multiplication to write S as 1/(N – 1) times the sum of N matrices of size p x p. For 1 ≤ k ≤ N, write X_k – M in place of  X̂_k.

Example 3

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n 1 Ν-Σ(-m)2 k=1