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Chapter 1: Problem 1

Compute the operator norm of the linear transformation defined by the following matrices: (a) \(\left[\begin{array}{rr}2 & 0 \\ 0 & -3\end{array}\right]\) (b) \(\left[\begin{array}{rr}1 & 2 \\ 0 & -1\end{array}\right]\) (c) \(\left[\begin{array}{ll}1 & 0 \\ 5 & 1\end{array}\right]\). Hint: In (c) maximize \(|A x|^{2}=26 x_{1}^{2}+10 x_{1} x_{2}+x_{2}^{2}\) subject to the constraint \(x_{1}^{2}+x_{2}^{2}=1\) and use the result of Problem 2 ; or use the fact that \(\|A\|=\left[\text { Max eigenvalue of } A^{T} A\right]^{1 / 2}\). Follow this same hint for \((b)\).

Short Answer

Expert verified
The operator norms are: (a) 3, (b) \( \sqrt{3 + 2\sqrt{2}} \), (c) \( \sqrt{26} \).

Step by step solution

01

Define the Operator Norm

The operator norm of a matrix \(A\) can be defined as \(\|A\| = \sqrt{\lambda_{\text{max}}(A^T A)}\), where \(\lambda_{\text{max}}\) is the maximum eigenvalue of \(A^T A\).
02

Compute \(A^T A\) for each matrix

For each given matrix \(A\), first compute \(A^T\) (the transpose of \(A\)), then compute \(A^T A\).
03

Calculate the eigenvalues of \(A^T A\)

Find the eigenvalues of each \(A^T A\) matrix calculated in Step 2. The operator norm will be the square root of the maximum eigenvalue.
04

Compute \(A^T A\) for matrix (a)

Given \(A = \left[ \begin{array}{rr}2 & 0 \ 0 & -3 \end{array} \right]\), we compute \(A^T = \left[ \begin{array}{rr}2 & 0 \ 0 & -3 \end{array} \right]\). Thus, \(A^T A = \left[ \begin{array}{rr} 2 & 0 \ 0 & -3 \end{array} \right] \left[ \begin{array}{rr}2 & 0 \ 0 & -3 \end{array} \right] = \left[ \begin{array}{rr} 4 & 0 \ 0 & 9 \end{array} \right]\).
05

Find the eigenvalues for matrix (a)

The eigenvalues of \( \left[ \begin{array}{rr} 4 & 0 \ 0 & 9 \end{array} \right]\) are \(4\) and \(9\). The maximum eigenvalue is \(9\).
06

Compute the operator norm for matrix (a)

The operator norm is \( \|A\| = \sqrt{9} = 3 \).
07

Compute \(A^T A\) for matrix (b)

Given \(A = \left[ \begin{array}{rr}1 & 2 \ 0 & -1 \end{array} \right]\), we compute \(A^T = \left[ \begin{array}{rr}1 & 0 \ 2 & -1 \end{array} \right]\). Thus, \(A^T A = \left[ \begin{array}{rr}1 & 2 \ 0 & -1 \end{array} \right]\left[ \begin{array}{rr}1 & 0 \ 2 & -1 \end{array} \right] = \left[ \begin{array}{rr} 5 & -2 \ -2 & 1 \end{array} \right]\).
08

Find the eigenvalues for matrix (b)

The characteristic polynomial of \( \left[ \begin{array}{rr} 5 & -2 \ -2 & 1 \end{array} \right]\) is \( \lambda^2 - 6\lambda + 1 = 0 \). The eigenvalues are \( 3 \pm 2\sqrt{2} \).
09

Compute the operator norm for matrix (b)

The maximum eigenvalue is \( 3 + 2\sqrt{2} \). Thus, the operator norm is \( \|A\| = \sqrt{3 + 2\sqrt{2}} \).
10

Compute \(A^T A\) for matrix (c)

Given \(A = \left[ \begin{array}{ll} 1 & 0 \ 5 & 1 \end{array} \right]\), we compute \(A^T = \left[ \begin{array}{ll} 1 & 5 \end{array} \ 0 & 1 \right]\). Thus, \(A^T A = \left[ \begin{array}{ll} 1 & 0 \ 5 & 1 \end{array} \right] \left[ \begin{array}{ll} 1 & 5 \ 0 & 1 \end{array} \right] = \left[ \begin{array}{ll} 1 & 5 \ 5 & 26 \end{array} \right]\).
11

Find the eigenvalues for matrix (c)

The characteristic polynomial of \( \left[ \begin{array}{ll} 1 & 5 \ 5 & 26 \end{array} \right]\) is \( \lambda^2 - 27\lambda + 1 = 0 \). The eigenvalues are \( 13.5 \pm 12.5 \), which is \(26\) and \(1\).
12

Compute the operator norm for matrix (c)

The maximum eigenvalue is \( 26 \). Thus, the operator norm is \( \|A\| = \sqrt{26} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Norm
We start by understanding what a matrix norm is. A matrix norm is a measure that assesses the 'size' or 'length' of a matrix. It helps in quantifying matrices and is used in various applications including computing stability of numerical algorithms and analyzing convergence of matrix series. One common type of matrix norm is the operator norm, which is defined using eigenvalues. It ensures consistent results across different mathematical operations.
Eigenvalues
Eigenvalues play a crucial role in many areas of linear algebra and its applications. They are values that give us deep insights into the properties of a matrix. To find eigenvalues, we must solve the characteristic equation \(\text{det}(A - \lambda I) = 0\), where \(A\) is our matrix and \(I\) is the identity matrix of the same dimension. Eigenvalues help in determining the behavior of linear transformations, such as scaling and rotating vectors. They are fundamental in finding the operator norm, as the operator norm is the square root of the maximum eigenvalue of \(A^T A\).
Linear Transformation
Linear transformations map one vector space to another, preserving vector addition and scalar multiplication. When using a matrix to represent a linear transformation, each column represents the image of each corresponding basis vector of the original vector space under the transformation. Linear transformations can include operations like rotations, translations, and scaling. Understanding linear transformations and their representations help us in comprehending more complex structures in higher dimensions.
Transpose Matrix
The transpose of a matrix, denoted as \(A^T\), is formed by swapping rows with columns. This means the element at position (i, j) in the original matrix A becomes the element at position (j, i) in \(A^T\). Transposing is particularly important in contexts like computing the operator norm, where it is essential to compute \(A^T A\). Transposed matrices are widely used in various mathematical computations and simplifications, especially in linear algebra where the relationships between vectors or equations frequently swap dimensions.

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Most popular questions from this chapter

Write the second-order differential equation $$ \ddot{x}+a x+b x=0 $$ as a system in \(\mathbf{R}^{2}\) and determine the nature of the equilibrium point at the origin.

Describe the separatrices for the linear system $$ \begin{aligned} &\dot{x}_{1}=x_{1}+2 x_{2} \\ &\dot{x}_{2}=3 x_{1}+4 x_{2} \end{aligned} $$ Hint: Find the eigenspaces for \(A\).

Let the \(n \times n\) matrix \(A\) have real, distinct eigenvalues. Let \(\phi\left(t, x_{0}\right)\) be the solution of the initial value problem $$ \begin{aligned} \dot{\mathbf{x}} &=A \mathbf{x} \\ \mathbf{x}(0) &=\mathbf{x}_{0} \end{aligned} $$ Show that for each fixed \(t \in \mathbf{R}\), $$ \lim _{\mathbf{y}_{0} \rightarrow \mathbf{x}_{0}} \phi\left(t, \mathrm{y}_{0}\right)=\phi\left(t, \mathrm{x}_{0}\right) $$ This shows that the solution \(\phi\left(t, x_{0}\right)\) is a continuous function of the initial condition.

What is the relationship between the vector fields defined by $$ \dot{\mathbf{x}}=A \mathbf{x} $$ and $$ \dot{\mathbf{x}}=k A \mathrm{x} $$ where \(k\) is a non-zero constant? (Describe this relationship both for \(k\) positive and \(k\) negative.)

Show that the operator norm of a linear transformation \(T\) on \(\mathbf{R}^{n}\) satisfies $$ \|T\|=\max _{|\mathbf{X}|=1}|T(\mathbf{x})|=\sup _{\mathbf{x} \neq 0} \frac{|T(\mathbf{x})|}{|\mathbf{x}|} $$
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