Description

Zhenye Na(zna2)
IE531: Algorithms for Data Analytics
1. Exercise 3.7 Let A be a square n×n matrix whose rows are orthonormal. Prove that the columns of A are orthonormal.
Solution:
Since the row vectors of A are orthonormal, we have that AA| = I. For square matrix A, this implies that A| = A−1 . Since A−1A = I, we have A|A = I, which implies that the column vectors of A are also orthonormal.
2. Exercise 3.13 Let Pσiuivi| be the singular value decomposition of a rank r matrix
i
k
A. Let Ak = P σiuivi| be a rank k approximation to A for some k < r. Express the
i=1
following quantities in terms of the singular values {σi,1 ≤ i ≤ r}.
• ||Ak||2F
k
Solution: P σi2
i=1

Solution:
• ||A − Ak||2F
r
Solution: P σi2 i k
Solution: σk2+1
3. Exercise 3.14 If A is a symmetric matrix with distinct singular values, show that the left and right singular vectors are the same and that A = V DV |.
Solution:
A. is a symmetric matrix, so that A|A = AA|. Suppose A = UDV | is the Singular Value Decomposition of A. Then
1
A|A = (UDV |)|UDV |
= V |D|U|UDV |
= V D2V |
AA| = UDV |(UDV |)|
= UD2U|
So the left and right singular vectors are the same and that A = V DV |.
4. Exercise 3.16 Use the power method to compute the singular value decomposition of the matrix.

Solution: Please see attachment.
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