1. Consider the case where Y is an n × p data matrix containing n observations on p-variables. The datamatrix has been demeaned. Use this information to answer the following questions. (4 marks).• S denotes the sample covariance matrix of Y. Using matrix notation, state the sample covariancematrix of Y. Be sure to clearly define any quantities needed to state S.• What is the dimension of the sample covariance matrix of Y.• What is the dimension of an arbitrary eigenvector associated with S?
Question
- Consider the case where Y is an n × p data matrix containing n observations on p-variables. The datamatrix has been demeaned. Use this information to answer the following questions. (4 marks).• S denotes the sample covariance matrix of Y. Using matrix notation, state the sample covariancematrix of Y. Be sure to clearly define any quantities needed to state S.• What is the dimension of the sample covariance matrix of Y.• What is the dimension of an arbitrary eigenvector associated with S?
Solution
• The sample covariance matrix S of Y is defined as S = (1/(n-1)) * Y' * Y, where Y' is the transpose of Y. This formula is derived from the definition of covariance, which measures the joint variability of two variables. In the context of a matrix, it measures the variability between each pair of variables (columns) in the matrix.
• The dimension of the sample covariance matrix S of Y is p × p. This is because the covariance matrix is a square matrix where each element represents the covariance between a pair of variables. Since there are p variables, there are p rows and p columns, hence the dimension is p × p.
• The dimension of an arbitrary eigenvector associated with S is p × 1. This is because an eigenvector of a matrix is a non-zero vector that only changes by a scalar factor when that matrix is multiplied by it. Since the matrix S is of dimension p × p, its eigenvectors are of dimension p × 1.
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