Show that A and $A^T$ have the same nonzero singular values. How are their singular value decompositions related?

Use the method of Example 1 to find the singular value decomposition of each of the following matrices:

1. $\left[\begin{array}{cc}1& 1\\ 2& 2\end{array}\right]$

2. $\left[\begin{array}{cc}2& -2\\ 1& 2\end{array}\right]$

3. $\left[\begin{array}{cc}1& 3\\ 3& 1\\ 0& 0\\ 0& 0\end{array}\right]$

4. $\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 1\\ 0& 1& 2\\ 0& 0& 0\end{array}\right]$

For each of the matrices in Exercise 2:

1. determine the rank.

2. find the closest (with respect to the Frobenius norm) matrix of rank 1.

Let

Find the closest (with respect to the Frobenius norm) matrices of rank 1 and rank 2 to A.

The matrix

has singular value decomposition

1. Use the singular value decomposition to find orthonormal bases for $R\left(A^T\right)$ and $N\left(A\right)$.

2. Use the singular value decomposition to find orthonormal bases for $R\left(A\right)$ and $N\left(A^T\right)$.

Prove that if A is a symmetric matrix with eigenvalues ${\lambda }_1,{\lambda }_2,\mathrm{.}\text{ . . },{\lambda }_n$, then the singular values of A are ${|\lambda }_1|,{|\lambda }_2|,\mathrm{.}\text{ . . },\left|{\lambda }_n\right|$.

Let A be an $m\times n$ matrix with singular value decomposition $U\mathrm{\sum }{V}^T$, and suppose that A has rank r, where $r<n$. Show that $\{{\text{v}}_1,\text{. . . },{\text{v}}_r\}$ is an orthonormal basis for $R\left(A^T\right)$.

Let A be an $n\times n$ matrix. Show that $A^{T}A$ and $A{A}^T$ are similar.

Let A be an $n\times n$ matrix with singular values ${\sigma }_1,{\sigma }_2,\mathrm{.}\text{ . . },{\sigma }_n$ and eigenvalues ${\lambda }_1,{\lambda }_2,\mathrm{.}\text{ . . },{\lambda }_n$. Show that

Let A be an $n\times n$ matrix with singular value decomposition $U\mathrm{\sum }{V}^T$ and let

Show that if

then the ${\mathbf{x}}_i$’s and ${\mathbf{y}}_i$’s are eigenvectors of B. How do the eigenvalues of B relate to the singular values of A?

Show that if $\sigma$ is a singular value of A, then there exists a nonzero vector x such that

Let A be an $m\times n$ matrix of rank n with singular value decomposition $U\mathrm{\sum }{V}^T$. Let ${\mathrm{\sum }}^{+}$ denote the $n\times m$ matrix

and define $A^{+}=V{\mathrm{\sum }}^{+}{U}^T$. Show that $\hat{x}=A{}^{+}\mathbf{b}$ satisfies the normal equations $A^{T}A\mathbf{x}=A^T\mathbf{b}$.

Let $A^{+}$ be defined as in Exercise 12 and let $P=A{A}^{+}$. Show that $P^2=P$ and $P^T=P$.