Estimate The Hilbert–Schmidt Norm Of $\mathrm E^{\mathrm ItA}-\mathrm E^{\mathrm ItB}$

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Introduction

In this article, we will delve into the estimation of the Hilbert-Schmidt norm of the expression $\mathrm e^{\mathrm itA}-\mathrm e^{\mathrm itB}$, where $A$ and $B$ are real-valued self-adjoint $(n \times n)$-matrices with operator norms less than or equal to 1. The Hilbert-Schmidt norm is a fundamental concept in functional analysis and operator theory, and its estimation is crucial in various applications, including signal processing and machine learning.

Preliminaries

Before we proceed with the estimation, let's recall some essential concepts and definitions.

Hilbert-Schmidt norm

The Hilbert-Schmidt norm of a matrix $M$ is defined as:

$$\|M\|_{HS} = \sqrt{\sum_{i,j} |M_{ij}|^2}$$

where $M_{ij}$ are the entries of the matrix $M$.

Operator norm

The operator norm of a matrix $M$ is defined as:

$$\|M\| = \sup_{\|x\|_2 = 1} \|Mx\|_2$$

where $\|x\|_2$ is the Euclidean norm of the vector $x$.

Self-adjoint matrices

A matrix $A$ is said to be self-adjoint if it satisfies the condition:

$$A^T = A$$

where $A^T$ is the transpose of the matrix $A$.

0-1 matrix

A matrix $D$ is said to be a 0-1 matrix if all its entries are either 0 or 1.

Estimation of the Hilbert-Schmidt norm

Now, let's proceed with the estimation of the Hilbert-Schmidt norm of the expression $\mathrm e^{\mathrm itA}-\mathrm e^{\mathrm itB}$.

Taylor expansion

We can expand the expression $\mathrm e^{\mathrm itA}-\mathrm e^{\mathrm itB}$ using the Taylor series expansion:

$$\mathrm e^{\mathrm itA}-\mathrm e^{\mathrm itB} = \sum_{k=0}^{\infty} \frac{(\mathrm itA)^k}{k!} - \sum_{k=0}^{\infty} \frac{(\mathrm itB)^k}{k!}$$

Hilbert-Schmidt norm of the Taylor expansion

We can estimate the Hilbert-Schmidt norm of the Taylor expansion as follows:

$$\|\mathrm e^{\mathrm itA}-\mathrm e^{\mathrm itB}\|_{HS} \leq \sum_{k=0}^{\infty} \frac{\|\mathrm itA\|^k}{k!} - \sum_{k=0}^{\infty} \frac{\|\mathrm itB\|^k}{k!}$$

Operator norm of the matrices

Since $A$ and $B$ are self-adjoint matrices with operator norms less than or equal to 1, we have:

$$\|\mathrm itA\| = \|\mathrm itB\| = \|\mathrm it\| \cdot \|A\| = \|\mathrm it\| \cdot \|B\| = 1$$

Estimation of the Hilbert-Schmidt norm

Substituting the operator norms of the matrices into the expression for the Hilbert-Schmidt norm, we get:

$$\|\mathrm e^{\mathrm itA}-\mathrm e^{\mathrm itB}\|_{HS} \leq \sum_{k=0}^{\infty} \frac{1}{k!} - \sum_{k=0}^{\infty} \frac{1}{k!}$$

Simplification

Simplifying the expression, we get:

$$\|\mathrm e^{\mathrm itA}-\mathrm e^{\mathrm itB}\|_{HS} \leq 0$$

Conclusion

In this article, we have estimated the Hilbert-Schmidt norm of the expression $\mathrm e^{\mathrm itA}-\mathrm e^{\mathrm itB}$, where $A$ and $B$ are real-valued self-adjoint $(n \times n)$-matrices with operator norms less than or equal to 1. We have shown that the Hilbert-Schmidt norm of the expression is less than or equal to 0.

Future work

In future work, we can explore the estimation of the Hilbert-Schmidt norm of more complex expressions involving matrices and operators. We can also investigate the applications of the Hilbert-Schmidt norm in various fields, including signal processing and machine learning.

References

• [1] Hilbert-Schmidt norm. In: Wikipedia, The Free Encyclopedia. Retrieved 2023-02-20.
• [2] Operator norm. In: Wikipedia, The Free Encyclopedia. Retrieved 2023-02-20.
• [3] Self-adjoint matrices. In: Wikipedia, The Free Encyclopedia. Retrieved 2023-02-20.
• [4] 0-1 matrix. In: Wikipedia, The Free Encyclopedia. Retrieved 2023-02-20.
  Q&A: Estimating the Hilbert-Schmidt norm of $\mathrm e^{\mathrm itA}-\mathrm e^{\mathrm itB}$ ===========================================================

Introduction

In our previous article, we estimated the Hilbert-Schmidt norm of the expression $\mathrm e^{\mathrm itA}-\mathrm e^{\mathrm itB}$, where $A$ and $B$ are real-valued self-adjoint $(n \times n)$-matrices with operator norms less than or equal to 1. In this article, we will answer some frequently asked questions related to the estimation of the Hilbert-Schmidt norm.

Q: What is the Hilbert-Schmidt norm?

A: The Hilbert-Schmidt norm of a matrix $M$ is defined as:

$$\|M\|_{HS} = \sqrt{\sum_{i,j} |M_{ij}|^2}$$

where $M_{ij}$ are the entries of the matrix $M$.

Q: What is the operator norm?

A: The operator norm of a matrix $M$ is defined as:

$$\|M\| = \sup_{\|x\|_2 = 1} \|Mx\|_2$$

where $\|x\|_2$ is the Euclidean norm of the vector $x$.

Q: What is a self-adjoint matrix?

A: A matrix $A$ is said to be self-adjoint if it satisfies the condition:

$$A^T = A$$

where $A^T$ is the transpose of the matrix $A$.

Q: What is a 0-1 matrix?

A: A matrix $D$ is said to be a 0-1 matrix if all its entries are either 0 or 1.

Q: How do you estimate the Hilbert-Schmidt norm of $\mathrm e^{\mathrm itA}-\mathrm e^{\mathrm itB}$?

A: We can estimate the Hilbert-Schmidt norm of the expression $\mathrm e^{\mathrm itA}-\mathrm e^{\mathrm itB}$ using the Taylor series expansion:

$$\mathrm e^{\mathrm itA}-\mathrm e^{\mathrm itB} = \sum_{k=0}^{\infty} \frac{(\mathrm itA)^k}{k!} - \sum_{k=0}^{\infty} \frac{(\mathrm itB)^k}{k!}$$

We can then estimate the Hilbert-Schmidt norm of the Taylor expansion as follows:

$$\|\mathrm e^{\mathrm itA}-\mathrm e^{\mathrm itB}\|_{HS} \leq \sum_{k=0}^{\infty} \frac{\|\mathrm itA\|^k}{k!} - \sum_{k=0}^{\infty} \frac{\|\mathrm itB\|^k}{k!}$$

Q: What is the significance of the operator norm of the matrices?

A: The operator norm of the matrices $A$ and $B$ is crucial in estimating the Hilbert-Schmidt norm of the expression $\mathrm e^{\mathrm itA}-\mathrm e^{\mathrm itB}$. Since $A$ and $B$ are self-adjoint matrices with operator norms less than or equal to 1, we have:

$$\|\mathrm itA\| = \|\mathrm itB\| = \|\mathrm it\| \cdot \|A\| = \|\mathrm it\| \cdot \|B\| = 1$$

Q: What is the final estimate of the Hilbert-Schmidt norm?

A: Substituting the operator norms of the matrices into the expression for the Hilbert-Schmidt norm, we get:

$$\|\mathrm e^{\mathrm itA}-\mathrm e^{\mathrm itB}\|_{HS} \leq \sum_{k=0}^{\infty} \frac{1}{k!} - \sum_{k=0}^{\infty} \frac{1}{k!}$$

Simplifying the expression, we get:

$$\|\mathrm e^{\mathrm itA}-\mathrm e^{\mathrm itB}\|_{HS} \leq 0$$

Conclusion

In this article, we have answered some frequently asked questions related to the estimation of the Hilbert-Schmidt norm of the expression $\mathrm e^{\mathrm itA}-\mathrm e^{\mathrm itB}$. We have shown that the Hilbert-Schmidt norm of the expression is less than or equal to 0.

Future work

In future work, we can explore the estimation of the Hilbert-Schmidt norm of more complex expressions involving matrices and operators. We can also investigate the applications of the Hilbert-Schmidt norm in various fields, including signal processing and machine learning.

References

• [1] Hilbert-Schmidt norm. In: Wikipedia, The Free Encyclopedia. Retrieved 2023-02-20.
• [2] Operator norm. In: Wikipedia, The Free Encyclopedia. Retrieved 2023-02-20.
• [3] Self-adjoint matrices. In: Wikipedia, The Free Encyclopedia. Retrieved 2023-02-20.
• [4] 0-1 matrix. In: Wikipedia, The Free Encyclopedia. Retrieved 2023-02-20.