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The resulting correlation matrix Ρ ρ ρ ρρ (8-15) is also the covariance matrix of the standardized variables. The matrix in (8-15) implies that the variables X1, X2,..., X, are equally correlated. It is not difficult to show (see Exercise 8.5) that the p eigenvalues of the corre- lation matrix (8-15) can be divided into two groups. When p is positive, the largest is λ == 1+ (p-1)p with associated eigenvector 1 1 1 ei = √p' √p The remaining p 1 eigenvalues are λ₂ = d3 = ... = and one choice for their eigenvectors is 1 -1 = λp = 1 - p e½ • - [VIX VIXX".")] 2 V1 2' ,0,...,0 (8-16) (8-17) 8.5. (a) Find the eigenvalues of the correlation matrix 1 ρ Ρ Ρ = ρ ρ ρ ρ 1 Are your results consistent with (8-16) and (8-17)? (b) Verify the eigenvalue-eigenvector pairs for the p xp matrix p given in (8-15).

The resulting correlation matrix Ρ ρ ρ ρρ (8-15) is also the covariance matrix of the standardized variables. The matrix in (8-15) implies that the variables X1, X2,..., X, are equally correlated. It is not difficult to show (see Exercise 8.5) that the p eigenvalues of the corre- lation matrix (8-15) can be divided into two groups. When p is positive, the largest is λ == 1+ (p-1)p with associated eigenvector 1 1 1 ei = √p' √p The remaining p 1 eigenvalues are λ₂ = d3 = ... = and one choice for their eigenvectors is 1 -1 = λp = 1 - p e½ • - [VIX VIXX".")] 2 V1 2' ,0,...,0 (8-16) (8-17) 8.5. (a) Find the eigenvalues of the correlation matrix 1 ρ Ρ Ρ = ρ ρ ρ ρ 1 Are your results consistent with (8-16) and (8-17)? (b) Verify the eigenvalue-eigenvector pairs for the p xp matrix p given in (8-15).

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter10: Matrices
Section10.CR: Chapter 10 Review
Problem 60CR
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The resulting correlation matrix Ρ ρ ρ ρρ (8-15) is also the covariance matrix of the standardized variables. The matrix in (8-15) implies that the variables X1, X2,..., X, are equally correlated. It is not difficult to show (see Exercise 8.5) that the p eigenvalues of the corre- lation matrix (8-15) can be divided into two groups. When p is positive, the largest is λ == 1+ (p-1)p with associated eigenvector 1 1 1 ei = √p' √p The remaining p 1 eigenvalues are λ₂ = d3 = ... = and one choice for their eigenvectors is 1 -1 = λp = 1 - p e½ • - [VIX VIXX".")] 2 V1 2' ,0,...,0 (8-16) (8-17)
Transcribed Image Text:The resulting correlation matrix Ρ ρ ρ ρρ (8-15) is also the covariance matrix of the standardized variables. The matrix in (8-15) implies that the variables X1, X2,..., X, are equally correlated. It is not difficult to show (see Exercise 8.5) that the p eigenvalues of the corre- lation matrix (8-15) can be divided into two groups. When p is positive, the largest is λ == 1+ (p-1)p with associated eigenvector 1 1 1 ei = √p' √p The remaining p 1 eigenvalues are λ₂ = d3 = ... = and one choice for their eigenvectors is 1 -1 = λp = 1 - p e½ • - [VIX VIXX".")] 2 V1 2' ,0,...,0 (8-16) (8-17)
8.5. (a) Find the eigenvalues of the correlation matrix 1 ρ Ρ Ρ = ρ ρ ρ ρ 1 Are your results consistent with (8-16) and (8-17)? (b) Verify the eigenvalue-eigenvector pairs for the p xp matrix p given in (8-15).
Transcribed Image Text:8.5. (a) Find the eigenvalues of the correlation matrix 1 ρ Ρ Ρ = ρ ρ ρ ρ 1 Are your results consistent with (8-16) and (8-17)? (b) Verify the eigenvalue-eigenvector pairs for the p xp matrix p given in (8-15).
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