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Unformatted text preview: One of the most useful result from linear algebra is the singular value decomposition theorem.
SVD Theorem: Let
matrices of size
entries, such that
Recall that a matrix
̅ be a matrix of size
and of size
of size ,
∑ to consider ̂
(2) Let , and assume that its rank is . Then there exist orthogonal
, and a diagonal matrix of size
with positive diagonal is said to be orthogonal if Form the matrix [ .. Define the mean vector ̅ to be In a lot of cases, we need to take the mean out of the data points which means we need
[ ̅ ̅]
̅ . Compute the product be the identity matrix of size (3) Show that ̂
(4) Let has a svd: What is the size of the resulting matrix? . Compute . (5) (continued from (4)). Let
corresponding columns of as given by the SVD Theorem above. Show that
. Then we can use the columns of to represent the
to achieve the dimension reduction. Show that and ...
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This note was uploaded on 01/16/2012 for the course MAD 4103 taught by Professor Li during the Spring '11 term at University of Central Florida.
- Spring '11