Exercise 5.3

Consider the matrix

(a)

Determine, on paper, a real SVD of $A$ in the form $A=U\mathrm{\Sigma }{V}^T$. The SVD is not unique, so find the one that has the minimal number of minus signs in $U$ and $V$.

(b)

List the singular values, left singular vectors, and right singular vectors of $A$. Draw a careful, labeled picture of the unit ball in ${ℝ}^2$ and its image under $A$, together with the singular vectors, with the coordinates of their vertices marked.

(c)

What are the $1$-, $2$-, $\infty$-, and Frobenius norms of $A$?

(d)

Find $A^{-1}$ not directly, but via the SVD.

(e)

Find the eigenvalues ${\lambda }_1,{\lambda }_2$ of $A$.

(f)

Verify that $\det <!--\; FUNCTION\; APPLICATION\; -->A={\lambda }_1{\lambda }_2$ and $|\det <!--\; FUNCTION\; APPLICATION\; -->A|={\sigma }_1{\sigma }_2$.

(g)

What is the area of the ellipsoid onto which $A$ maps the unit ball of ${ℝ}^2$?