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Chapter 7: Problem 15

Show that the energy norm is indeed a norm when the associated matrix is symmetric positive definite.

Short Answer

Expert verified
Answer: Yes, the energy norm is a norm when the associated matrix is symmetric positive definite. This is proven by demonstrating that the energy norm satisfies the three properties of a norm, including the positivity, linearity, and the triangle inequality.

Step by step solution

01

Definition of Energy Norm and Symmetric Positive Definite Matrix

First, let's recall the definition of the energy norm associated with a symmetric positive definite matrix A. An energy norm is defined as: \[ ||x||_A = \sqrt{(x^T A x)} \] A matrix A is symmetric positive definite if it satisfies: 1. A = A^T (symmetric) 2. x^T A x > 0 for all non-zero vectors x (positive definite)
02

Property 1: Proving Positivity

We need to show that the energy norm is always non-negative and is equal to zero if and only if x is the zero vector. Since A is symmetric positive definite, \[ x^T A x > 0 \] for all non-zero vectors x. Therefore, \[ ||x||_A = \sqrt{(x^T A x)} \ge 0 \] If x is the zero vector, \[ ||x||_A = \sqrt{(0^T A 0)} = 0 \] If ||x||_A = 0, it is clear that x=0, since \[x^T A x = 0 \] only if x is the zero vector. Hence, the energy norm satisfies the positivity property of a norm.
03

Property 2: Proving Linearity

We need to show that for any scalar 伪 and any vector x, ||伪x||_A = |伪| * ||x||_A. \[ ||\alpha x||_A = \sqrt{((\alpha x)^T A (\alpha x))} = \sqrt{(\alpha^2 x^T A x)} = |\alpha| \sqrt{(x^T A x)} = |\alpha| ||x||_A \] Thus, the energy norm satisfies the linearity property of a norm.
04

Property 3: Proving Triangle Inequality

We need to show that for any two vectors x and y, ||x + y||_A 鈮� ||x||_A + ||y||_A. This can be shown by applying the Cauchy-Schwarz inequality which states that for any vectors x and y, \[ (x^T y)^2 \le (x^T x)(y^T y) \] Applying the inequality, we get \[ ||x + y||_A^2 = (x + y)^T A (x + y) = x^T A x + 2x^T A y + y^T A y \le x^T A x + 2 |x^T A y| + y^T A y \] Using Cauchy-Schwarz inequality with A^{1/2}x and A^{1/2}y, we have \[ (x^T A y)^2 \le (x^T A x)(y^T A y) \] Now, we need to show that \[ \sqrt{x^T A x + 2 |x^T A y| + y^T A y} \le \sqrt{x^T A x} + \sqrt{y^T A y} \] Square both sides of the inequality: \[ x^T A x + 2 |x^T A y| + y^T A y \le x^T A x + 2 \sqrt{(x^T A x)(y^T A y)} + y^T A y \] This inequality holds based on our application of the Cauchy-Schwarz inequality on the left side terms. Hence, \[ ||x + y||_A \le ||x||_A + ||y||_A \] So, the energy norm satisfies the triangle inequality property of a norm. By showing that the energy norm satisfies the properties of positivity, linearity, and the triangle inequality, we have demonstrated that the energy norm is indeed a norm when the associated matrix is symmetric positive definite.

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Key Concepts

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

symmetric positive definite matrix
A symmetric positive definite matrix, often abbreviated as SPD matrix, is an essential concept in linear algebra that is widely used in calculus and numerical computations. To fully understand this concept, it's important to break down its properties:
  • Symmetry: A matrix \( A \) is symmetric if it is equal to its transpose, which means \( A = A^T \). This property ensures that the matrix behaves uniformly in all directions.

  • Positive Definiteness: A matrix \( A \) is positive definite if \( x^T A x > 0 \) for all non-zero vectors \( x \). This property guarantees that the quadratic form \( x^T A x \) always yields a positive value, except when the vector is the zero vector.

SPD matrices are crucial as they ensure that certain transformations preserve the geometry of spaces in a meaningful way. They provide a firm ground for defining norms, particularly the energy norm, as they give room for ensuring that the quadratic form always remains positive, leading to meaningful norm calculations.
Cauchy-Schwarz inequality
The Cauchy-Schwarz inequality is a cornerstone in the study of vector spaces, providing a fundamental property that relates to the inner product of vectors. It's expressed as follows:
  • For any two vectors \( x \) and \( y \) in an inner product space, \( (x^T y)^2 \le (x^T x)(y^T y) \).

This inequality tells us that the square of the dot product of two vectors is less than or equal to the product of their magnitudes. It's similar to the geometrical interpretation that the angle between two vectors affects their dot product.

In the context of norms, the Cauchy-Schwarz inequality helps in proving the triangle inequality property. By guaranteeing an upper bound on the product \( |x^T A y| \), we can assert control over combined vector magnitudes, a critical step when proving certain properties of the energy norm.
triangle inequality
The triangle inequality is an easily remembered concept derived from the physical properties of triangles in geometry. In the context of norms, it provides a meaningful way to understand distances in vector spaces. Specifically, for any vectors \( x \) and \( y \), the triangle inequality states:
  • \( ||x + y||_A \le ||x||_A + ||y||_A \)

This property indicates that the direct path (a single vector) to a point is always shorter or equal to the path which takes a detour (sum of vectors).

In proving this property for the energy norm, combining the positive definiteness of the matrix with the Cauchy-Schwarz inequality plays a key role. This ensures that no unexpected behavior appeared when adding vectors, such as longer-than-expected vector lengths, thus maintaining the robustness of the norm structure.
norm properties
Norms are a fundamental concept in the study of vector spaces, quantifying the "size" or "length" of vectors. A well-defined norm must satisfy the following three properties:
  • Non-negativity: For any vector \( x \), \( ||x|| \ge 0 \), and \( ||x|| = 0 \) if and only if \( x \) is the zero vector.

  • Scalability: For any scalar \( \alpha \) and vector \( x \), \( ||\alpha x|| = |\alpha| ||x|| \). This property ensures that scaling a vector increases or decreases its norm proportionally.

  • Triangle Inequality: For any vectors \( x \) and \( y \), the sum of their norms satisfies \( ||x + y|| \le ||x|| + ||y|| \).

In the context of the energy norm, these properties ensure its validity as a norm. The energy norm specifically utilizes the properties of symmetric positive definite matrices to satisfy these criteria, creating a robust framework for analyzing vector lengths in spaces where the geometry is governed by the specific SPD matrix in question.

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Most popular questions from this chapter

(a) Write a program for solving the linear least squares problems that arise throughout the iterations of the GMRES method, using Givens rotations, where the matrix is a nonsquare \((k+1) \times k\) upper Hessenberg matrix. Specifically, solve $$ \min _{\mathbf{z}}\left\|\rho e_{1}-H_{k+1, k} \mathbf{z}\right\| $$ Provide a detailed explanation of how this is done in your program, and state what \(Q\) and \(R\) in the associated QR factorization are. (b) Given \(H_{k+1, k}\), suppose we perform a step of the Arnoldi process. As a result, we now have a new upper Hessenberg matrix \(H_{k+2, k+1}\). Describe the relationship between the old and the new upper Hessenberg matrices and explain how this relationship can be used to solve the new least squares problem in an economical fashion. (c) The least squares problems throughout the iterations can be solved using a QR decomposition approach. Show that the upper triangular factor cannot be singular unless \(\mathbf{x}_{k}=\mathbf{x}\), the exact solution.

Write a program that solves the problem of Example \(7.13\) for \(N=127\) and \(N=255\) using a multigrid method. The script in that example and the code for a V-cycle in Section \(7.6\) should prove useful with appropriate modification. Set your coarsest level (where the problem is solved exactly, say, by a direct solver) to \(N=31\). You should be able to obtain considerably faster convergence than in Example \(7.13\) at the price of a considerable headache.

\(x\) $$ A=\left(\begin{array}{cc} 2 & -1 \\ -1 & 2 \end{array}\right) $$ and suppose we are required to solve \(A \mathbf{x}=\mathbf{b}\). (a) Write down explicitly the iteration matrices corresponding to the Jacobi, Gauss-Seidel, and SOR schemes. (b) Find the spectral radius of the Jacobi and Gauss-Seidel iteration matrices and the asymptotic rates of convergence for these two schemes. (c) Plot a graph of the spectral radius of the SOR iteration matrix vs. the relaxation parameter \(\omega\) for \(0 \leq \omega \leq 2\). (d) Find the optimal SOR parameter, \(\omega^{*}\). What is the spectral radius of the corresponding iteration matrix? Approximately how much faster would SOR with \(\omega^{*}\) converge compared to Jacobi?

Suppose CG is applied to a symmetric positive definite linear system \(A \mathbf{x}=\mathbf{b}\) where the righthand-side vector \(\mathbf{b}\) happens to be an eigenvector of the matrix \(A\). How many iterations will it take to converge to the solution? Does your answer change if \(A\) is not SPD and instead of CG we apply GMRES?

Consider the saddle point linear system $$ \underbrace{\left(\begin{array}{cc} A & B^{T} \\ B & 0 \end{array}\right)}_{\mathcal{K}}\left(\begin{array}{l} u \\ p \end{array}\right)=\left(\begin{array}{l} f \\ g \end{array}\right), $$ where \(A\) is \(n \times n\), symmetric positive definite and \(B\) is \(m \times n\) with \(m
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