Question

Let A∈Rᵐˣⁿ be a full rank matrix with m≥n. Such matrices were the object of focus at the end of Unit 3 , when we solved least-squares problems. In that unit, we introduced two ways of solving these problems: (1) using the normal equation and (2) using the QR equation. I claim we can use the SVD to solve least-squares problems, too. (a) (3 points) By starting with the normal equation AᵀAx=Aᵀb, use the reduced SVD of A to show that x=VΣ⁻¹Uᵀb Justify each step of your calculation. Be careful: the matrix U is not quite an orthogonal matrix (as it isn't square in general). You'll have to think carefully about the product UᵀU. (b) (3 points) Suppose you are given the following least-squares problem: (1−1​)x=(20​). Solve this problem using your result from (a).

Answer 1

Using the Singular Value Decomposition method, the question demonstrates solving a least-squares problem by converting the normal equation into a solution for x using the components of SVD, then applies this approach to a specific problem with matrix A and vector b.

Step-by-step explanation:

The question involves solving a least-squares problem using the Singular Value Decomposition (SVD) method. Assuming a full rank matrix A from R⁽¹⁴¹, we typically solve least-squares problems by leveraging the normal equation AᵀAx=Aᵀb. Through the SVD of A, we can express the solution as x=VΣ⁻¹Uᵀb.

Given the specific least-squares problem (1 -1)x=(2 0), we can use the derived SVD result to find the solution for x. First, we need to compute the SVD of matrix A which is UΣVᵀ then substitute back into x=VΣ⁻¹Uᵀb to get the values of x.