Problem 22 A weighted $\ell_{\infty}$ -no... [FREE SOLUTION]

Chapter 4: Problem 22

A weighted $\ell_{\infty}$ -norm is a norm on $\mathbb{R}^{n}$ of the form $$ \|x\|=\max _{1 \leq i \leq n} w_{i}\left|x_{i}\right| $$ where $w_{1}, w_{2}, \ldots, w_{n}$ are fixed positive numbers called weights. Prove the norm postulates for this norm. What is the subordinate matrix norm?

Short Answer

Expert verified

The weighted $ \ell_\infty $ norm satisfies positivity, homogeneity, and the triangle inequality. The subordinate matrix norm is $ \|A\| = \max_{1 \leq i \leq m} \sum_{j=1}^n w_j \left| a_{ij} \right| $.

Step by step solution

Norm Positivity and Definiteness

To show the norm positivity and definiteness, we observe that for any vector $ x \in \mathbb{R}^n $, $ \left| x_i \right| $ is non-negative for all $ i $, and the weights $ w_i $ are given to be positive. Therefore, $ w_i \left| x_i \right| \geq 0 $ for each component, and hence $ \|x\| = \max_{1 \leq i \leq n} w_i \left| x_i \right| \geq 0 $. Additionally, $ \|x\| = 0 \implies w_i \left| x_i \right| = 0 $ for all $ i $, which implies that $ x_i = 0 $ for each $ i $, and so $ x = 0 $. Thus, the norm is positive definite.

Homogeneity

For homogeneity, we need to show that $ \| \alpha x \| = |\alpha| \cdot \| x \| $ for any scalar $ \alpha $ and vector $ x $. We have $ \| \alpha x \| = \max_{1 \leq i \leq n} w_i \left| \alpha x_i \right| = \max_{1 \leq i \leq n} w_i |\alpha| \left| x_i \right| = |\alpha| \cdot \max_{1 \leq i \leq n} w_i \left| x_i \right| = |\alpha| \cdot \| x \| $. Thus, homogeneity is verified.

Triangle Inequality

To show the triangle inequality, we need $ \|x + y\| \leq \|x\| + \|y\| $ for vectors $ x, y \in \mathbb{R}^n $. Consider $ \|x + y\| = \max_{1 \leq i \leq n} w_i \left| x_i + y_i \right| \leq \max_{1 \leq i \leq n} w_i \left(\left| x_i \right| + \left| y_i \right|\right) \leq \max_{1 \leq i \leq n} w_i \left| x_i \right| + \max_{1 \leq i \leq n} w_i \left| y_i \right| = \|x\| + \|y\| $. This confirms that the triangle inequality holds.

Identify Subordinate Matrix Norm

The subordinate matrix norm for the $ \ell_\infty $ norm is defined for a matrix $ A = [a_{ij}] \in \mathbb{R}^{m \times n} $ as \[ \|A\| = \max_{1 \leq i \leq m} \sum_{j=1}^n w_j \left| a_{ij} \right|. \] This norm reflects the maximum weighted sum of the absolute values of the elements in any row of the matrix, considering the weights provided by the $ \ell_\infty $ vector norm.

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Key Concepts

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

Weighted norms

In mathematics, weighted norms are a type of norm that involves scaling vector components with prescribed positive constants called weights. Consider a vector in $ \mathbb{R}^n $ with a weighted $ \ell_\infty $ norm described as follows:

For a vector $ x $, the norm is calculated as $ \|x\| = \max_{1 \leq i \leq n} w_i |x_i| $.
The weights $ w_i $ are positive numbers that adjust the influence of each component $ x_i $ on the norm.
These weights are crucial in giving importance to certain elements of the vector over others.

This kind of norm is particularly useful in various applications, like optimization, where some components of a vector have more significance than others. By assigning different weights, we can analyze the vector's magnitude while paying more attention to higher-weighted components.

Subordinate matrix norm

The subordinate matrix norm is a derived norm for matrices based on vector norms. Specifically, the matrix norm inherited from a weighted $ \ell_\infty $ norm is defined as follows:

For a matrix $ A = [a_{ij}] $ in $ \mathbb{R}^{m \times n} $, its subordinate matrix norm is given by $ \|A\| = \max_{1 \leq i \leq m} \sum_{j=1}^n w_j |a_{ij}| $.
This formula calculates the maximum weighted sum of absolute values for each row in the matrix.
The norm measures the largest possible effect of the matrix acting on a vector in terms of weighted influence.

Subordinate norms like this are valuable in understanding how matrices transform or enlarge vectors, particularly when each vector component has particular importance defined by weights. It's a critical tool for evaluating matrix behaviors, especially in fields like numerical analysis and linear algebra.

Triangle inequality

The triangle inequality is a fundamental property that every norm in mathematics must satisfy. For vector norms, it implies the following relationship:

For any vectors $ x $ and $ y $ in $ \mathbb{R}^n $, the inequality $ \|x + y\| \leq \|x\| + \|y\| $ holds.
In the context of the weighted $ \ell_\infty $ norm, this means the maximum weighted absolute value attained by the sum of two vectors does not exceed the individual sums.
Geometrically, it ensures that the direct path between two points in space is no longer than a detour path through an additional point.

The triangle inequality offers a means to understand the relationships and distances between vector additions, ensuring consistency and predictability in length or magnitude measurements.

Homogeneity

Homogeneity is another essential property of norms that describes how the norm scales with respect to scalar multiplication:

For any vector $ x $ in $ \mathbb{R}^n $ and any scalar $ \alpha $, the homogeneity property asserts that $ \| \alpha x \| = |\alpha| \cdot \|x\| $.
In simpler terms, multiplying a vector by a scalar should scale the norm of the vector by the absolute value of that scalar.
This property ensures that the norm is consistent with scalar multiplication, reflecting proportionate changes in vector size accurately.

In practical terms, homogeneity maintains that if the weights or influences of individual components of a vector change linearly (increased or decreased), the overall measure of the vector's magnitude changes in a predictable and proportional fashion.

Norm positivity

Norm positivity is a crucial aspect of norm definitions in mathematics, ensuring a basic but fundamental property:

A vector norm $ \|x\| $ is always non-negative, meaning $ \|x\| \geq 0 $ for any vector $ x $ in $ \mathbb{R}^n $.
The norm becomes zero if and only if the vector itself is zero, indicated by $ x = 0 $.
This property guarantees that norms are true measures of size, with no possibility of negative values, thus preserving mathematical integrity in calculations.

In weighted norms, even with varying component weights, the positivity condition remains intact because the weights are also positive. This ensures reliability and consistency when assessing vector magnitudes, regardless of individual weights.

91影视

A weighted \(\ell_{\infty}\) -norm is a norm on \(\mathbb{R}^{n}\) of the form $$ \|x\|=\max _{1 \leq i \leq n} w_{i}\left|x_{i}\right| $$ where \(w_{1}, w_{2}, \ldots, w_{n}\) are fixed positive numbers called weights. Prove the norm postulates for this norm. What is the subordinate matrix norm?

Short Answer

Step by step solution

Norm Positivity and Definiteness

Homogeneity

Triangle Inequality

Identify Subordinate Matrix Norm

Key Concepts

Weighted norms

Subordinate matrix norm

Triangle inequality

Homogeneity

Norm positivity

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