Determine $\Vert {\cdot \Vert }_F$, $\Vert {\cdot \Vert }_{\infty }$, and $\Vert {\cdot \Vert }_1$ for each of the following matrices:

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

2. $\left[\begin{array}{cc}\hfill 1& 4\\ \hfill -2& 2\end{array}\right]$

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

4. $\left[\begin{array}{ccc}0& 5& 1\\ 2& 3& 1\\ 1& 2& 2\end{array}\right]$

5. $\left[\begin{array}{ccc}5& 0& 5\\ 4& 1& 0\\ 3& 2& 1\end{array}\right]$

Let

and set

Determine the value of $\Vert {A\Vert }_2$ by finding the maximum value of f for all $(x_1,x_2)\ne (0,0)$.

Let

Use the method of Exercise 2 to determine the value of ${\Vert A\Vert }_2$

Let

1. Compute the singular value decomposition of D.

2. Find the value of ${\Vert D\Vert }_2$.

Show that if D is an $n\times n$ diagonal matrix, then

If D is an $n\times n$ diagonal matrix, how do the values of ${\Vert D\Vert }_1$, ${\Vert D\Vert }_2$, and ${\Vert D\Vert }_{\infty }$ compare? Explain your answers.

Let I denote the $n\times n$ identity matrix. Determine the values of ${\Vert I\Vert }_1$, ${\Vert I\Vert }_{\infty }$, and ${\Vert I\Vert }_F$.

Let ${\Vert \cdot \Vert }_M$ denote a matrix norm on ${ℝ}^{n\times n}$, ${\Vert \cdot \Vert }_V$ denote a vector norm on ${ℝ}^n$, and I be the $n\times n$ identity matrix. Show that

1. if $\Vert {\cdot \Vert }_M$ and $\Vert {\cdot \Vert }_V$ are compatible, then $\Vert {I\Vert }_M\ge 1$.

2. if $\Vert {\cdot \Vert }_M$ is subordinate to $\Vert {\cdot \Vert }_V$, then $\Vert I{\Vert }_M=1$.

A vector x in ${ℝ}^n$ can also be viewed as an $n\times 1$ matrix X:

1. How do the matrix norm ${\Vert X\Vert }_{\infty }$ and the vector norm ${\Vert \mathbf{x}\Vert }_{\infty }$ compare? Explain.

2. How do the matrix norm ${\Vert X\Vert }_1$ and the vector norm ${\Vert \mathbf{x}\Vert }_1$ compare? Explain.

A vector y in ${ℝ}^n$ can also be viewed as an $n\times 1$ matrix $Y=\left(\mathbf{\text{y}}\right)$. Show that

1. ${\Vert Y\Vert }_2=\Vert \mathbf{\text{y}}{\Vert }_2$

2. ${\Vert Y^T\Vert }_2=\Vert \mathbf{\text{y}}{\Vert }_2$

Let $A={\mathbf{\text{wy}}}^T$, where $\mathbf{\text{w}}\in {ℝ}^m$ and $\mathbf{\text{y}}\in {ℝ}^n$. Show that

1. $\frac{{\displaystyle {\Vert A\mathbf{x}\Vert }_2}}{{\displaystyle \Vert {\mathbf{x}\Vert }_2}}\le {\Vert \text{y}\Vert }_2{\Vert \mathbf{w}\Vert }_2\text{ for all }\mathbf{x}\ne \mathbf{0}\text{ in }{ℝ}^n$.

2. ${\Vert A\Vert }_2=\mathbf{\Vert }{\mathbf{y}\mathbf{\Vert }}_{\mathbf{2}}\Vert {\mathbf{w}\Vert }_2$

Let

1. Determine ${\Vert A\Vert }_{\infty }$.

2. Find a vector x whose coordinates are each $\pm 1$ such that ${\Vert A\mathbf{x}\Vert }_{\infty }={\Vert A\Vert }_{\infty }$. (Note that ${\Vert \mathbf{x}\Vert }_{\infty }=1$, so ${\Vert A\Vert }_{\infty }={\Vert A\mathbf{x}\Vert }_{\infty }/{\Vert \mathbf{x}\Vert }_{\infty }$.)

Theorem 7.4.2 states that

Prove this in two steps.

1. Show first that

   $${\Vert A\Vert }_{\infty }\le \underset{1\le i\le m}{\text{max}}\left(\underset{j=1}{\overset{n}{\mathrm{\Sigma }}}|a_{ij}|\right)$$

2. Construct a vector x whose coordinates are each $\pm 1$ such that

   $$\frac{{\displaystyle {\Vert A\mathbf{x}\Vert }_{\infty }}}{{\displaystyle {\Vert \mathbf{x}\Vert }_{\infty }}}={\Vert A\mathbf{x}\Vert }_{\infty }=\underset{1\le i\le m}{\text{max}}\left(\underset{j=1}{\overset{n}{\mathrm{\Sigma }}}\left|a_{ij}\right|\right)$$

Show that ${\Vert A\Vert }_F={\Vert A^T\Vert }_F$.

Let A be a symmetric $n\times n$ matrix. Show that ${\Vert A\Vert }_{\infty }={\Vert A\Vert }_1$.

Let A be a $5\times 4$ matrix with singular values ${\sigma }_1=5$, ${\sigma }_2=3$, and ${\sigma }_3={\sigma }_4=1$. Determine the values of ${\Vert A\Vert }_2$ and ${\Vert A\Vert }_F$.

Let A be an $m\times n$ matrix.

1. Show that ${\Vert A\Vert }_2\le {\Vert A\Vert }_F$.

2. Under what circumstances will ${\Vert A\Vert }_2={\Vert A\Vert }_F$?

Let $\Vert \cdot \Vert$ denote a family of vector norms and let ${\Vert \cdot \Vert }_M$ be a subordinate matrix norm. Show that

Let A be an $m\times n$ matrix and let $\Vert {\cdot \Vert }_V$ and $\Vert {\cdot \Vert }_W$ be vector norms on ${ℝ}^n$ and ${ℝ}^m$, respectively. Show that

defines a matrix norm on ${ℝ}^{m\times n}$.

Let A be an $m\times n$ matrix. The (1,2)-norm of A is given by

(See Exercise 19.) Show that

Let A be an $m\times n$ matrix. Show that ${\Vert A\Vert }_{(1,2)}\le {\Vert A\Vert }_2$.

Let $A\in {ℝ}^{m\times n}$ and $B\in {ℝ}^{n\times r}$. Show that

1. ${\Vert A\mathbf{x}\Vert }_2\le {\Vert A\Vert }_{(1,2)}{\Vert \mathbf{x}\Vert }_1$ for all x in ${ℝ}^n$.

2. ${\Vert AB\Vert }_{(1,2)}\le {\Vert A\Vert }_2{\Vert B\Vert }_{(1,2)}$

3. ${\Vert AB\Vert }_{(1,2)}\le {\Vert A\Vert }_{(1,2)}{\Vert B\Vert }_1$

Let A be an $n\times n$ matrix and let ${\Vert \cdot \Vert }_M$ be a matrix norm that is compatible with some vector norm on ${ℝ}^n$. Show that if $\lambda$ is an eigenvalue of A, then $\left|\lambda \right|\le {\Vert A\Vert }_M$.

Use the result from Exercise 23 to show that if $\lambda$ is an eigenvalue of a stochastic matrix, then $\left|\lambda \right|\le 1$.

Sudoku is a popular puzzle involving matrices. In this puzzle, one is given some of the entries of a $9\times 9$ matrix A and asked to fill in the missing entries. The matrix A has a block structure

where each submatrix $A_{ij}$ is $3\times 3$. The rules of the puzzle are that each row, each column, and each of the submatrices of A must be made up of all of the integers 1 through 9. We will refer to such a matrix as a sudoku matrix. Show that if A is a sudoku matrix, then $\lambda =45$ is its dominant eigenvalue.

Let $A_{ij}$ be a submatrix of a sudoku matrix A (see Exercise 25). Show that if $\lambda$ is an eigenvalue of $A_{ij}$, then $\left|\lambda \right|\le 22$.

Let A be an $n\times n$ matrix and $\mathbf{x}\in {ℝ}^n$. Prove:

1. ${\Vert A\mathbf{x}\Vert }_{\infty }\le n^{1/2}{\Vert A\Vert }_2\Vert {\mathbf{x}\Vert }_{\infty }$

2. ${\Vert A\mathbf{x}\Vert }_2\le n^{1/2}{\Vert A\Vert }_{\infty }\Vert {\mathbf{x}\Vert }_2$

3. $n^{-1/2}{\Vert A\Vert }_2\le {\Vert A\Vert }_{\infty }\le n^{1/2}{\Vert A\Vert }_2$

Let A be a symmetric $n\times n$ matrix with eigenvalues ${\lambda }_1,\text{. . .},{\lambda }_n$ and orthonormal eigenvectors ${\mathbf{u}}_1,\text{. . . },{\mathbf{u}}_n$. Let $\mathbf{x}\in {ℝ}^n$ and let $c_i={\mathbf{u}}_i^T\mathbf{x}$ for $i=1,2,\text{. . .}\text{ , }n$. Show that

1. ${\Vert A\mathbf{x}\Vert }_2^2=\underset{i=1}{\overset{n}{\mathrm{\Sigma }}}{\left({\lambda }_i{c}_i\right)}^2$

2. if $\mathbf{x}\mathrm{\ne }\mathbf{0}$, then

   $$\underset{1\le i\le n}{\text{min}}\left|{\lambda }_i\right|\le \frac{{\displaystyle {\Vert A\mathbf{x}\Vert }_2}}{{\displaystyle {\Vert \mathbf{x}\Vert }_2}}\le \underset{1\le i\le n}{\text{max}}|{\lambda }_i|$$

3. ${\Vert A\Vert }_2=\underset{1\le i\le n}{\text{max}}\left|{\lambda }_i\right|$

Let

Find $A^{-1}$ and ${\text{cond}}_{\infty }\left(A\right)$.

Solve the given two systems and compare the solutions. Are the coefficient matrices well conditioned? Ill conditioned? Explain.

Let

Calculate ${\text{cond}}_{\infty }\left(A\right)={\Vert A\Vert }_{\infty }{\Vert A^{-1}\Vert }_{\infty }$.

Let A be a nonsingular $n\times n$ matrix, and let ${\Vert \cdot \Vert }_M$ denote a matrix norm that is compatible with some vector norm on ${ℝ}^n$. Show that

Let

for each positive integer n. Calculate

1. $A_n^{-1}$

2. ${\text{cond}}_{\infty }\left(A_n\right)$

3. $\underset{n\to \infty }{\text{lim}}{\text{ cond}}_{\infty }\left(A_n\right)$

If A is a $5\times 3$ matrix with ${\Vert A\Vert }_2=8$, ${\text{cond}}_2\left(A\right)=2$, and ${\Vert A\Vert }_F=12$, determine the singular values of A.

Given

if two-digit decimal floating-point arithmetic is used to solve the system $A\mathbf{x}=\mathbf{b}$, the computed solution will be $\mathbf{x}={(1.1,0.88)}^T$.

1. Determine the residual vector r and the value of the relative residual ${\Vert \mathbf{r}\Vert }_{\infty }/{\Vert \mathbf{b}\Vert }_{\infty }$.

2. Find the value of ${\text{cond}}_{\infty }\left(A\right)$.

3. Without computing the exact solution, use the results from parts (a) and (b) to obtain bounds for the relative error in the computed solution.

4. Compute the exact solution x and determine the actual relative error. Compare this to the bounds derived in part (c).

Let

Calculate ${\text{cond}}_1\left(A\right)={\Vert A\Vert }_1\Vert {A^{-1}\Vert }_1$.

Let A be the matrix in Exercise 36 and let

Let x and x′ be the solutions of $A\mathbf{x}=\mathbf{b}$ and $A^{\prime }\mathbf{x}=\mathbf{b}$, respectively, for some $\mathbf{b}\in {ℝ}^3$. Find a bound for the relative error $\left(\Vert {\mathbf{x}-{\mathbf{x}}^{\prime }\Vert }_1\right)/\Vert {{\mathbf{x}}^{\prime }\Vert }_1$.

Let

An approximate solution of $A\mathbf{x}=\mathbf{b}$ is calculated by rounding the entries of b to the nearest integer and then solving the rounded system with integer arithmetic. The calculated solution is ${\text{x}}^{\prime }={(12,4,2,1)}^T$. Let r denote the residual vector.

1. Determine the values of ${\Vert \mathbf{r}\Vert }_{\infty }$ and ${\text{cond}}_{\infty }\left(A\right)$.

2. Use your answer to part (a) to find an upper bound for the relative error in the solution.

3. Compute the exact solution x and determine the relative error $\frac{{\displaystyle {\Vert \mathbf{x}-{\mathbf{x}}^{\prime }\Vert }_{\infty }}}{{\displaystyle {\Vert \mathbf{x}\Vert }_{\infty }}}$.

Let A and B be nonsingular $n\times n$ matrices. Show that

Let D be a nonsingular $n\times n$ diagonal matrix and let

1. Show that

   $${\text{cond}}_1\left(D\right)={\text{cond}}_{\infty }\left(D\right)=\frac{{\displaystyle d_{\max }}}{{\displaystyle d_{\min }}}$$

2. Show that

   $${\text{cond}}_2\left(D\right)=\frac{{\displaystyle d_{\max }}}{{\displaystyle d_{\min }}}$$

Let Q be an $n\times n$ orthogonal matrix. Show that

1. ${\Vert Q\Vert }_2=1$

2. ${\text{cond}}_2\left(Q\right)=1$

3. for any $\mathbf{b}\in {ℝ}^n$, the relative error in the solution of $Q\mathbf{x}=\mathbf{b}$ is equal to the relative residual, that is,

   $$\frac{{\displaystyle {\Vert \mathbf{e}\Vert }_2}}{{\displaystyle {\Vert \mathbf{x}\Vert }_2}}=\frac{{\displaystyle {\Vert \mathbf{\text{r}}\Vert }_2}}{{\displaystyle {\Vert \mathbf{b}\Vert }_2}}$$

Let A be an $n\times n$ matrix and let Q and V be $n\times n$ orthogonal matrices. Show that

1. ${\Vert QA\Vert }_2={\Vert A\Vert }_2$

2. ${\Vert AV\Vert }_2={\Vert A\Vert }_2$

3. ${\Vert QAV\Vert }_2={\Vert A\Vert }_2$

Let A be an $m\times n$ matrix and let ${\sigma }_1$ be the largest singular value of A. Show that if x and y are nonzero vectors in ${ℝ}^n$, then each of the following holds:

1. $\frac{{\displaystyle \left|{\mathbf{x}}^{T}A\mathbf{y}\right|}}{{\displaystyle \Vert {\mathbf{x}\Vert }_2\Vert {\mathbf{y}\Vert }_2}}\le {\sigma }_1$

   [Hint: Make use of the Cauchy–Schwarz inequality.]

2. $\underset{\mathbf{x}\ne \mathbf{0},\mathbf{y}\ne \mathbf{0}}{\text{max}}\frac{{\displaystyle \left|{\mathbf{x}}^{T}A\mathbf{y}\right|}}{{\displaystyle \Vert \mathbf{x}\Vert \Vert \mathbf{y}\Vert }}={\sigma }_1$

Let A be an $m\times n$ matrix with singular value decomposition $U\mathrm{\Sigma }{V}^T$. Show that

Let A be an $m\times n$ matrix with singular value decomposition $U\mathrm{\Sigma }{V}^T$. Show that, for any vector $\mathbf{x}\in {ℝ}^n$,

Let A be a nonsingular $n\times n$ matrix and let Q be an $n\times n$ orthogonal matrix. Show that

1. ${\text{cond}}_2\left(QA\right)={\text{cond}}_2\left(AQ\right)={\text{cond}}_2\left(A\right)$

2. if $B=Q^{T}AQ$, then ${\text{cond}}_2\left(B\right)={\text{cond}}_2\left(A\right)$.

Let A be a symmetric nonsingular $n\times n$ matrix with eigenvalues ${\lambda }_1,\text{. . . },{\lambda }_n$. Show that