Let ¤1,..., IN be data in R" with sample covariance C = XXT. Suppose u; is an eigenvector of C corresponding to the eigenvalues di, with A, > d2 >...Am 2 0. Let P; be the orthogonal projection onto the Span({u;}). N-1 (i) Show that the sample variance of P;x1,., P;xN_ is d;. 1
Let ¤1,..., IN be data in R" with sample covariance C = XXT. Suppose u; is an eigenvector of C corresponding to the eigenvalues di, with A, > d2 >...Am 2 0. Let P; be the orthogonal projection onto the Span({u;}). N-1 (i) Show that the sample variance of P;x1,., P;xN_ is d;. 1
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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please send handwritten solution for part ii
![Q3. Let x1,..., XN be data in R" with sample covariance C = ,XXT. Suppose u; is an
eigenvector of C corresponding to the eigenvalues di, with A1 2 d2 ...Am 2 0. Let
P; be the orthogonal projection onto the Span({u;}).
(i) :
Show that the sample variance of P;x1,... , P;xN is X;.
1
(ii)
Suppose r < i <m is the rank of C. What is the sample variance? Explain
and interpret.
(ii)
Let d be an integer and let Ua be the matrix of orthonormal eigenvectors
[u1,..., ua] for C.
Suppose now that x; = TN + UaUF (x; – TN) + Zi, for all 1 <i < N. Our model
assumes that the Z;'s are independent samples from N(0, o²I,m) for an unknown
variance o?. Write down the MLE for o² and express it in terms of the eigenvalues
of C.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3fd61ef9-3587-4735-a2c1-cfaec8f950c1%2F1160279e-a3df-4d89-b5cc-37517d90fadb%2Fdjte4g8_processed.png&w=3840&q=75)
Transcribed Image Text:Q3. Let x1,..., XN be data in R" with sample covariance C = ,XXT. Suppose u; is an
eigenvector of C corresponding to the eigenvalues di, with A1 2 d2 ...Am 2 0. Let
P; be the orthogonal projection onto the Span({u;}).
(i) :
Show that the sample variance of P;x1,... , P;xN is X;.
1
(ii)
Suppose r < i <m is the rank of C. What is the sample variance? Explain
and interpret.
(ii)
Let d be an integer and let Ua be the matrix of orthonormal eigenvectors
[u1,..., ua] for C.
Suppose now that x; = TN + UaUF (x; – TN) + Zi, for all 1 <i < N. Our model
assumes that the Z;'s are independent samples from N(0, o²I,m) for an unknown
variance o?. Write down the MLE for o² and express it in terms of the eigenvalues
of C.
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