## Description

General Instructions

The following document contains the solutions to the theory-based questions for

Question 1

Finding the singular values of

ATA

− λI| = 10 200 − λ 120

λ = 332.17,90.50,0.33

Thus, the largest and smallest singular values of the matrix are and

Similarly for B

BTB

− λI| = 40 800 − λ 480

λ = 1328.68,361.99,1.33

Thus, the largest and smallest singular values of the matrix are and

Note that B = 2A and the eigenvalues of B are 4 times the eigenvalues of A.

Thus we can conclude that κ(αA) = κ(A) but under the added constraint of α ∈ R∖{0}

Question 2

The transformation matrix A corresponding to the rotation of a vector x by an angle θ is given by

cosθ −sinθ

A = [ ] sinθ cosθ

Computing the singular values of A by computing the eigenvalues of ATA

T cosθ sinθ cosθ −sinθ

A A = [ ][ ]

−sinθ cosθ sinθ cosθ cos2 θ + sin2 θ −cosθsinθ + sinθcosθ

=−sinθcosθ + cosθsinθ sin2 θ + cos2 θ ]

1 0

= [ ]

0 1

The eigenvalues of ATA are λ1 = 1 and λ2 = 1. Thus, the singular values of A are σ1 = 1 and σ2 = 1.

Thus, the condition number of A is given by

κ(A) = ||σσ12|| = 11 = 1

Question 3

Finding the transformation matrix A corresponding to the reflection of a vector x about a given vector v.

Let θ be the angle between x and v given by .

For reflection about v, we need to rotate x by 2θ. Thus, the transformation matrix A is given by

cos2θ

A = [

sin2θ −sin2θ] cos2θ

As we calculated in the previous question the singular values of the above matrix are σ1 = 1 and σ2 = 1 (as they don’t depend on θ) Thus, the condition number of A is given by

κ(A) = ||σσ12|| = 11 = 1

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