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Linear Algebra With Applications
Found in: Page 413
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

Consider a singular value decomposition A=UΣVTof an n×mmatrix A with rankA = m. Let u1,,unbe the columns of U . Without using the results of Chapter 5 , computeA(ATA)-1ATui. Explain your result in terms of Theorem 5.4.7.

AATA-1ATu˙i=u˙i if 1im0 otherwise

See the step by step solution

Step by Step Solution

Step 1:To compute AATA-1ATu→i.

Let's first compute ATu˙i.


whereei is the ith basis vector of n.

Since ΣT is an n×m matrix

role="math" localid="1660679956475" ΣTei=σiei if 1im0 otherwise


role="math" localid="1660680127331" V ΣTei=V σiei =σiv˙i if 1im0 otherwise


ATu˙i=σiv˙i if 1im0 otherwise

Step2: ToExplain the result in terms of Theorem 5.4.7.

Recall that v˙i's are eigenvectors of ATA with eigenvalues σi2.

Hence, ATAv˙i=σi2v˙i.This implies that ATA-1σiv˙i=1σiv˙i.

This implies the following.

ATA-1ATu˙i=1σiv˙i if 1im0 otherwise

Now, use that



AATA-1ATu˙i=u˙i if 1im0 otherwise

If we are considering the sub space of, say W, which is spanned byu˙1,u˙2,,u˙m , then the above result implies that the projection u˙ionto W is u˙iif 1imand is 0 if m+1in.

Step 3:Final proof

AATA-1ATu˙i=u˙i if 1im0 otherwise

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