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Chapter 7: Problem 13

Find the norms and condition numbers from the square roots of \(\lambda_{\max }\left(A^{\top} A\right)\) and \(\lambda_{\min }\left(A^{\mathrm{T}} A\right)\) $$ \left(\left[\begin{array}{rr} -2 & 0 \\ 0 & 2 \end{array}\right) \quad\left[\begin{array}{ll} 1 & 1 \\ 0 & 0 \end{array}\right] \quad\left[\begin{array}{rr} 1 & 1 \\ -1 & 1 \end{array}\right]\right. $$

Short Answer

Expert verified
Norms: 2, 鈭�2, 鈭�2; Condition Numbers: 1, 鈭�, 1.

Step by step solution

01

Understand the Definitions

To find the norms and condition numbers, we need to calculate the singular values of each matrix, which are the square roots of the eigenvalues of \(A^{\top}A\). The largest singular value corresponds to the matrix norm \( \|A\| \) and the condition number is given by \( \frac{\lambda_{\max}(A^\top A)}{\lambda_{\min}(A^\top A)} \).We need to evaluate\( \lambda_{\max}(A^\top A) \) and \( \lambda_{\min}(A^\top A) \) for each of the given matrices.
02

Calculate \(A^{\top} A\) for each matrix

For each provided matrix \( A \), calculate the matrix product \( A^{\top} A \). 1. For \( A_1 = \begin{bmatrix} -2 & 0 \ 0 & 2 \end{bmatrix} \): \( A_1^{\top} A_1 = \begin{bmatrix} -2 & 0 \ 0 & 2 \end{bmatrix}\begin{bmatrix} -2 & 0 \ 0 & 2 \end{bmatrix} = \begin{bmatrix} 4 & 0 \ 0 & 4 \end{bmatrix} \)2. For \( A_2 = \begin{bmatrix} 1 & 1 \ 0 & 0 \end{bmatrix} \): \( A_2^{\top} A_2 = \begin{bmatrix} 1 & 0 \ 1 & 0 \end{bmatrix}\begin{bmatrix} 1 & 1 \ 0 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} \)3. For \( A_3 = \begin{bmatrix} 1 & 1 \ -1 & 1 \end{bmatrix} \): \( A_3^{\top} A_3 = \begin{bmatrix} 1 & -1 \ 1 & 1 \end{bmatrix}\begin{bmatrix} 1 & 1 \ -1 & 1 \end{bmatrix} = \begin{bmatrix} 2 & 0 \ 0 & 2 \end{bmatrix} \)
03

Compute Eigenvalues of \(A^{\top} A\)

Find the eigenvalues for each computed matrix \( A^{\top} A \).1. For \( \begin{bmatrix} 4 & 0 \ 0 & 4 \end{bmatrix} \), the eigenvalues are \( \lambda_1 = 4 \) and \( \lambda_2 = 4 \).2. For \( \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} \), the eigenvalues are \( \lambda_1 = 2 \) and \( \lambda_2 = 0 \).3. For \( \begin{bmatrix} 2 & 0 \ 0 & 2 \end{bmatrix} \), the eigenvalues are \( \lambda_1 = 2 \) and \( \lambda_2 = 2 \).
04

Compute Norms and Condition Numbers

Using the eigenvalues, compute norms and condition numbers.1. For \( A_1 \): - Norm \( \|A_1\| = \sqrt{\lambda_{\max}} = \sqrt{4} = 2 \) - Condition Number \( \kappa(A_1) = \frac{4}{4} = 1 \)2. For \( A_2 \): - Norm \( \|A_2\| = \sqrt{\lambda_{\max}} = \sqrt{2} \approx 1.41 \) - Condition Number \( \kappa(A_2) = \frac{2}{0} = \infty \)3. For \( A_3 \): - Norm \( \|A_3\| = \sqrt{\lambda_{\max}} = \sqrt{2} \approx 1.41 \) - Condition Number \( \kappa(A_3) = \frac{2}{2} = 1 \)

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Key Concepts

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

Matrix Norms
Matrix norms provide a way to measure the size or length of a matrix, much like vector norms do for vectors. The most commonly used matrix norm is the spectral norm, which is the largest singular value of the matrix. For a matrix \( A \), this norm is given by \( \|A\| = \sqrt{\lambda_{\max}(A^\top A)} \), where \( \lambda_{\max} \) denotes the largest eigenvalue of \( A^\top A \).
For the matrices provided:
  • Matrix \( A_1 \) has eigenvalues of \( 4 \) and thus its norm is \( 2 \).
  • Matrix \( A_2 \) has a largest eigenvalue of \( 2 \) leading to a norm of approximately \( 1.41 \).
  • Matrix \( A_3 \) also results in a norm of approximately \( 1.41 \).
The matrix norm is vital for various calculations in numerical methods, stability analysis, and controlling the error bounds in approximations.
Singular Values
Singular values are a set of non-negative numbers typically used to measure the significance or strength of the dimensions along which a matrix can be transformed. They are derived from the eigenvalues of the matrix \( A^\top A \), known as \( \sigma_i = \sqrt{\lambda_i} \). These values are key to understanding how a matrix scales and transform spaces.
For each given matrix:
  • Matrix \( A_1 \) possesses singular values of \( 2 \) because the eigenvalues are \( 4 \).
  • Matrix \( A_2 \), having eigenvalues \( 2 \) and \( 0 \), has singular values of approximately \( 1.41 \) and \( 0 \).
  • Matrix \( A_3 \) holds singular values again of approximately \( 1.41 \), matching its calculations of the eigenvalues.
Singular values are essential in many applications, such as signal processing, image compression, and machine learning for reducing dimensions while preserving critical information.
Condition Number
The condition number of a matrix is a measure of how sensitive the solution of a system of linear equations is to changes in the data. It is calculated as the ratio of the largest to the smallest singular value. The condition number, \( \kappa(A) \), is given by \( \frac{\sigma_{\max}}{\sigma_{\min}} \), and offers insights into numerical stability and error propagation.
For the given matrices:
  • Matrix \( A_1 \) has a condition number of \( 1 \), indicating stability since the singular values are equal.
  • Matrix \( A_2 \) has an infinite condition number suggesting that it is singular or poorly conditioned, which is due to a singular value of \( 0 \).
  • Matrix \( A_3 \) also shows a stable condition number of \( 1 \), similar to \( A_1 \).
A condition number close to \( 1 \) typically indicates that the matrix is well-conditioned and less sensitive to errors, whereas higher numbers can indicate numerical instability in computational tasks.
Eigenvalues
Eigenvalues are fundamental components in linear algebra that reveal quite a bit about a matrix's characteristics, acting as scaling factors for the corresponding eigenvectors. For a matrix \( A \), the eigenvalues are obtained by solving the equation \( \det(A - \lambda I) = 0 \). These values help in determining many matrix properties, including norms and conditions discussed above.
In this exercise, calculating \( A^\top A \) for each matrix:
  • Matrix \( A_1 \) shows eigenvalues as \( 4 \) and \( 4 \), leading to equal scaling in all directions.
  • Matrix \( A_2 \) possesses eigenvalues \( 2 \) and \( 0 \), highlighting a non-invertible or degenerate transformation where direction corresponding to zero is squished to zero.
  • Matrix \( A_3 \) possesses eigenvalues \( 2 \) and \( 2 \), indicating a uniform scaling independent of direction.
Eigenvalues are used extensively in stability analysis of systems, differential equations, and even in quantum mechanics, providing insight into the intrinsic properties of a matrix.

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Most popular questions from this chapter

When \(A\) is multiplied by \(P_{i j}\) (plane rotation), which entries are changed? When \(P_{i j} A\) is multiplied on the right by \(P_{i j}^{-1}\), which entries are changed now?

Find the \(L U\) factorization of \(A=\left[\begin{array}{ll}6 & 1 \\ 1 & 1\end{array}\right]\). On your computer, solve by elimination when \(\epsilon=10^{-3}, 10^{-6}, 10^{-9}, 10^{-12}, 10^{-15}\) $$ \left[\begin{array}{ll} \epsilon & 1 \\ 1 & 1 \end{array}\right]\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]=\left[\begin{array}{c} 1+\epsilon \\ 2 \end{array}\right] $$ The true \(x\) is \((1,1)\). Make a table to show the error for each \(\epsilon\). Exchange the two equations and solve again-the errors should almost disappear.

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Comparing the eigenvalues of \(A^{T} A\) and \(A A^{\top}\), prove that \(\|A\|=\left\|A^{\top}\right\|\).
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