2. In this question, you will explore the geometry behind SVD. Let A = 3√3 35 3√3 (a) Find an SVD decomposition of A (b) Draw a picture of what happens to the square S with vertices (0, 0), (1, 0), (1, 1), (0, 1) under the transformation A. (c) There is an angle a € [0, 2π] such that if we rotate S by a degrees (counterclockwise starting from the positive x-axis), then the image of the rotated square under A becomes a rectangle, which we denote by R. Find a and draw a picture of R (make sure its vertices are clearly labeled). (d) What is the angle (counterclockwise with respect to the positive x-axis) one needs to rotate the rectangle R (from the previous part) by so that its legs will lie on the x and y-axis? (e) Explain in words the relation between the SVD decomposition and the answers to part (c) and (d)
2. In this question, you will explore the geometry behind SVD. Let A = 3√3 35 3√3 (a) Find an SVD decomposition of A (b) Draw a picture of what happens to the square S with vertices (0, 0), (1, 0), (1, 1), (0, 1) under the transformation A. (c) There is an angle a € [0, 2π] such that if we rotate S by a degrees (counterclockwise starting from the positive x-axis), then the image of the rotated square under A becomes a rectangle, which we denote by R. Find a and draw a picture of R (make sure its vertices are clearly labeled). (d) What is the angle (counterclockwise with respect to the positive x-axis) one needs to rotate the rectangle R (from the previous part) by so that its legs will lie on the x and y-axis? (e) Explain in words the relation between the SVD decomposition and the answers to part (c) and (d)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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question (b), (c), (d) and (e) please, thanks!
![### Exploring the Geometry Behind Singular Value Decomposition (SVD)
Let the matrix \( A \) be defined as:
\[
A = \begin{pmatrix} \frac{3\sqrt{3}}{4} & \frac{1}{4} \\ -\frac{5}{4} & \frac{3\sqrt{3}}{4} \end{pmatrix}
\]
#### (a) Finding the SVD of \( A \)
Find the Singular Value Decomposition (SVD) of matrix \( A \).
#### (b) Transformation of Square \( S \)
Draw a picture to show the effect of the transformation \( A \) on the square \( S \) with vertices \((0,0)\), \((1,0)\), \((1,1)\), \((0,1)\).
#### (c) Rotation and the Transformation to Rectangle \( R \)
Find an angle \( \alpha \in [0, 2\pi] \) such that rotating \( S \) by \( \alpha \) degrees counterclockwise turns its image under transformation \( A \) into a rectangle \( R \). Draw the rectangle \( R \) and ensure its vertices are labeled.
#### (d) Alignment of Rectangle \( R \)
Determine the angle (counterclockwise in relation to the positive \( x \)-axis) required to rotate rectangle \( R \) so that its sides are parallel to the \( x \)- and \( y \)-axes.
#### (e) Relation Between SVD and Geometry
Explain the relationship between the SVD decomposition and your answers to parts (c) and (d).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fafdd1b11-86dc-46df-9f91-cc7e20db01fb%2F5dffed65-69cf-4399-bcc5-790c4b05dcb5%2Fw9pb4ay_processed.png&w=3840&q=75)
Transcribed Image Text:### Exploring the Geometry Behind Singular Value Decomposition (SVD)
Let the matrix \( A \) be defined as:
\[
A = \begin{pmatrix} \frac{3\sqrt{3}}{4} & \frac{1}{4} \\ -\frac{5}{4} & \frac{3\sqrt{3}}{4} \end{pmatrix}
\]
#### (a) Finding the SVD of \( A \)
Find the Singular Value Decomposition (SVD) of matrix \( A \).
#### (b) Transformation of Square \( S \)
Draw a picture to show the effect of the transformation \( A \) on the square \( S \) with vertices \((0,0)\), \((1,0)\), \((1,1)\), \((0,1)\).
#### (c) Rotation and the Transformation to Rectangle \( R \)
Find an angle \( \alpha \in [0, 2\pi] \) such that rotating \( S \) by \( \alpha \) degrees counterclockwise turns its image under transformation \( A \) into a rectangle \( R \). Draw the rectangle \( R \) and ensure its vertices are labeled.
#### (d) Alignment of Rectangle \( R \)
Determine the angle (counterclockwise in relation to the positive \( x \)-axis) required to rotate rectangle \( R \) so that its sides are parallel to the \( x \)- and \( y \)-axes.
#### (e) Relation Between SVD and Geometry
Explain the relationship between the SVD decomposition and your answers to parts (c) and (d).
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