Consider a singular value decomposition of an matrix A with rank. Let be the columns of U . Without using the results of Chapter 5 , compute Explain your result in terms of Theorem 5.4.7.
Let's first compute
where is the basis vector of
Since is an matrix
Recall that are eigenvectors of with eigenvalues
Hence, This implies that
This implies the following.
Now, use that
If we are considering the sub space of, say W, which is spanned by , then the above result implies that the projection onto W is if and is 0 if
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