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## 15.10 Problems

15.1 Prove that for all rank-one matrices, $σ12=∑i=1m∑j=1naij2$. Hint: Use Theorem 15.5.

15.2 Suppose u1, u2, …, un and v1, v2, …, vn are orthonormal bases for Ρn. Construct the matrix A that transforms each vi into ui to give Av1 = u1, Av2 = u2, …, Avn = un.

15.3 Let $A=[ a11a120a22 ]$. Determine value(s) for the aij so that A has distinct singular values.

15.4 Prove that $rank(ATA)=rank(AAT)$.

15.5 Find the SVD of

a. ATA.

b. $(ATA)−1$

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