MATH 475 - Spring 2022 – Homework 1
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MATH 475 - Spring 2022 – Homework 1
Questions
1. Let A V Rm×n . Show that
m
|A|1 = max |αij|﹐
n
|A|& = max |αij|.
2. Prove Lemmas 29 and 30 of Set 1 of the Lecture Notes.
3. Let A V Rm×n and # = minem﹐ n}. Show that |A|F = ′σ1(2) +...+ σp(2), where σ 1 ﹐...﹐ σp are the singular values of A.
4. Let A V Rm×n , where m > n, have the SVD A = I ΣIT . Derive an expression for the SVD T of the matrix = (AT A)– 1 AT. Simplify you answer as much as possible. What are the sizes of the matrices , and ?
5. Let A V Rm×m . Show that |det(A)| = σi, where σ 1 ﹐...﹐ σm are the singular values
of A.
2022-03-14