Question

Let \(\mathbf{A}=\left[a_{i j}\right]\) be an \(m \times n\) matrix and $$ \lambda=\max \left\\{\left|a_{i j}\right| \mid 1 \leq i \leq m, 1 \leq i \leq n\right\\} $$ Show that \(\|\mathbf{A}\| \leq \lambda \sqrt{m n}\)

Short answer

Question: Prove that for a matrix A of size m x n, its Frobenius norm ||A||, is less than or equal to λ√(mn), where λ is the maximum absolute value of all the elements of the matrix A. Answer: To prove this inequality, we define the Frobenius norm as ||A|| = √(Σ a_{ij}²), where a_{ij} are the elements of matrix A. Since |a_{ij}| ≤ λ, we have a_{ij}² ≤ λ². Summing up these inequalities for all indices i and j, we get Σ a_{ij}² ≤ mnλ². Taking the square root of both sides and applying the definition of the Frobenius norm, we obtain ||A|| ≤ λ√(mn).

Step-by-step solution

Definition of the Frobenius norm

The Frobenius norm of a matrix A, denoted as ||A||, is defined as the square root of the sum of the squares of all its elements. Mathematically, this can be written as: $$ ||A|| = \sqrt{\sum_{i=1}^m \sum_{j=1}^n a_{ij}^2} $$

Calculate the sum of squares of all the elements of A

We know that each element of A, \(a_{ij}\), is less than or equal to the maximum element λ. So, we can write \(|a_{ij}| \leq \lambda\) for all \(1 \leq i \leq m\) and \(1 \leq j \leq n\). Squaring both sides, we get \(a_{ij}^2 \leq \lambda^2\). Now let's sum up these inequalities for all \(i\) and \(j\): $$ \sum_{i=1}^m \sum_{j=1}^n a_{ij}^2 \leq \sum_{i=1}^m \sum_{j=1}^n \lambda^2 $$ Note that the right side of the inequality is the sum of mn identical terms \(\lambda^2\). So, we can rewrite this as: $$ \sum_{i=1}^m \sum_{j=1}^n a_{ij}^2 \leq m n \lambda^2 $$

Take the square root of both sides and apply the definition of the Frobenius norm

Now, we take the square root of both sides of the inequality: $$ \sqrt{\sum_{i=1}^m \sum_{j=1}^n a_{ij}^2} \leq \sqrt{m n \lambda^2} $$ From the definition of the Frobenius norm, the left side of the inequality is equal to ||A||. Also, the right side can be simplified as \(\lambda\sqrt{mn}\): $$ ||A|| \leq \lambda \sqrt{mn} $$ We have successfully proven the inequality for the Frobenius norm of the matrix A.

Key concepts

Matrix Analysis

Matrices are fundamental in mathematics since they organize numbers in rows and columns, forming a rectangular array. Understanding matrices involves not only recognizing their size, denoted by their dimensions (e.g., an \(m \times n\) matrix), but also systematically analyzing their components and operations.When looking into matrix properties, one essential aspect is the matrix norm. Specifically, the Frobenius norm gives us a measure of the magnitude of a matrix by considering all its elements. It's calculated as the square root of the sum of the squares of each element in the matrix, which involves:

• Calculating each element's square.
• Summing all these squared values.
• Taking the square root of this total sum.

The Frobenius norm, and norms in general, provide critical insights into the behavior and characteristics of matrices under various transformations.

Inequalities

Inequalities are mathematical expressions defining the relative size or order between two objects. They play a crucial role in various fields like optimization, calculus, and of course, matrix analysis.In matrix terms, an inequality can compare norms, which gives us bounds on what the norms can be, providing a measure of predictability and control over the matrix properties. Specifically, the inequality \(||\mathbf{A}|| \leq \lambda \sqrt{mn}\) helps us understand and assure that the Frobenius norm of matrix \(\mathbf{A}\) won't exceed a particular limit, introducing:

• A comparison of each matrix element with the maximum element \(\lambda\).
• Bounding the sum of squares of matrix elements by \(mn\lambda^2\).

This bounding technique gives a clear indication of how changes or manipulations of the matrix can potentially affect its magnitude.

Real Analysis

Real analysis is the branch of mathematics that deals with real numbers and real-valued sequences and functions. It underpins many aspects of calculus and provides a deep understanding of limits, continuity, and series.In the context of matrices, real analysis principles are applied to ensure that operations such as norm calculations are well-founded. For example,

• Analyzing the convergence of series made from matrix element squares.
• Applying limits to make valid analytical conclusions about matrix properties.

The maximum element \(\lambda\) and the concept of bounding ensure a rigorous mathematical framework that helps verify inequalities like the one surrounding the Frobenius norm. Such analysis allows us to validate results through logical reasoning and strict mathematical rules, enhancing our understanding of matrices in a deeply analytical way.