Problems

01

Computing SVD

Compute an SVD of the matrix $A=\left(\begin{array}{rr}7& 1\\ 5& 5\\ 0& 0\end{array}\right)$.

02

SVD of $A^{\mathsf{T}}$

• (a) Given an SVD $A=U\Sigma V^{\mathsf{T}}$, what is a formula for an SVD of $A^{\mathsf{T}}$? What are the singular values of $A^{\mathsf{T}}$?
• (b) Compute an SVD of the matrix $A=\left(\begin{array}{rrr}3& 2& 2\\ 2& 3& -2\end{array}\right)$ by first computing an SVD of $A^{\mathsf{T}}$ and then applying your result in (a).

03

Optimizing $|A\mathbf{x}|$ using singular values

• (a) Verify that $|A\mathbf{x}{|}^2=Q(\mathbf{x})$ where $Q(\mathbf{x})$ is the quadratic form of $A^{\mathsf{T}}A$.
• (b) Suppose $A=\left(\begin{array}{rr}4& 6\\ 0& 4\end{array}\right)$. Starting with the observation in (a), find a unit vector $\mathbf{x}$ that maximizes the value of $|A\mathbf{x}|$ (and hence of $|A\mathbf{x}{|}^2)$. Describe your answer in terms of singular values and singular vectors of $A$.

04

Optimizing $|A\mathbf{x}|$ to find singular vectors

Show that the second largest singular value of $A$ is equal to the maximum of $|A\mathbf{x}|$ where $\mathbf{x}$ varies over all unit vectors which are orthogonal to $\mathbf{v}$, where $\mathbf{v}$ is a right singular vector corresponding to the largest singular value of $A$.

(Notation clarification: if the singular values are listed in order of size ${\sigma }_1\ge {\sigma }_2\ge {\sigma }_3\ge \cdots \ge {\sigma }_{\ell }$, then the largest singular value is ${\sigma }_1$, the second largest is ${\sigma }_2$, and the right singular vector in the problem is ${\mathbf{v}}_1$ satisfying $A{\mathbf{v}}_1={\sigma }_1{\mathbf{u}}_1$.)

05

Computing SVD

Compute an SVD of the matrix $A=\left(\begin{array}{rrrc}-18& 13& -4& 4\\ 2& 19& -4& 12\\ -14& 11& -12& 8\\ -2& 21& 4& 8\end{array}\right)$. You may use a calculator or computer, but you must show the steps of computation as in the example of computing an SVD by hand.