Problem 8 Zeigen Sie, daB die folgenden Ma... [FREE SOLUTION] (2024)

Get started for free

Log In Start studying!

Get started for free Log out

Chapter 1: Problem 8

Zeigen Sie, daB die folgenden Matrizen unitär sind: $$ \begin{aligned} &\mathbf{A}=\frac{1}{\sqrt{12}}\left(\begin{array}{ccc} 2 & 1-\sqrt{3} j & -1-\sqrt{3} j \\ -2 j & \sqrt{3}-j & \sqrt{3}+j \\ -2 & 2 & -2 \end{array}\right) \\ &B=\frac{1}{\sqrt{3}}\left(\begin{array}{cc} 1+j & 1 \\ j & -1-j \end{array}\right), \quad C=\left(\begin{array}{ll} j & 0 \\ 0 & j \end{array}\right) \end{aligned} $$

Short Answer

Expert verified

Die Matrizen \( \mathbf{A} \), \( \mathbf{B} \) und \( \mathbf{C} \) sind unitär.

Step by step solution

01

Definition of Unitär Matrix

Eine Matrix ist unitär, wenn sie invertierbar ist und ihr Inversens gleich ihrem Adjungiertens ist, das heißt, für eine Matrix \( U \) gilt: \( U^{-1} = U^* \). Insbesondere gilt \( U^* U = I \), wobei \( U^* \) die konjugiert transponierte Matrix von \( U \) ist und \( I \) die Einheitsmatrix ist.

02

Konjugiert Transponierte berechnen für \( \mathbf{A} \)

Finde die konjugiert transponierte Matrix \( \mathbf{A}^* \) von \( \mathbf{A} \). Für \( \mathbf{A} \) ist \[ \mathbf{A}^* = \frac{1}{\sqrt{12}} \begin{pmatrix} 2 & 2j & -2\ 1+\sqrt{3}j & \sqrt{3}+j & 2\ -1+\sqrt{3}j & \sqrt{3}-j & -2 \end{pmatrix} \]

03

Überprüfen Sie \( \mathbf{A}^* \mathbf{A} \)=\( \mathbf{I} \)

Multiplizieren \( \mathbf{A} \) und \( \mathbf{A}^* \) : \[ \mathbf{A} \cdot \mathbf{A}^* = \frac{1}{12} \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix} = \mathbf{I} \] Dies beweist, dass \( \mathbf{A} \) unitär ist.

04

Konjugiert Transponierte berechnen für \( \mathbf{B} \)

Finde die konjugiert transponierte Matrix \( \mathbf{B}^* \) von \( \mathbf{B} \). Für \( \mathbf{B} \) ist \[ \mathbf{B}^* = \frac{1}{\sqrt{3}} \begin{pmatrix} 1-j & -j \ 1 & -1+j \end{pmatrix} \]

05

Überprüfen Sie \( \mathbf{B}^* \mathbf{B} \)=\( \mathbf{I} \)

Multiplizieren \( \mathbf{B} \) und \( \mathbf{B}^* \): \[ \mathbf{B} \cdot \mathbf{B}^* = \frac{1}{3} \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} = \mathbf{I} \] Dies beweist, dass \( \mathbf{B} \) unitär ist.

06

Konjugiert Transponierte berechnen für \( \mathbf{C} \)

Finde die konjugiert transponierte Matrix \( \mathbf{C}^* \) von \( \mathbf{C} \). Für \( \mathbf{C} \) ist \[ \mathbf{C}^* = \begin{pmatrix} -j & 0 \ 0 & -j \end{pmatrix} \]

07

Überprüfen Sie \( \mathbf{C}^* \mathbf{C} \)=\( \mathbf{I} \)

Multiplizieren \( \mathbf{C} \) und \( \mathbf{C}^* \): \[ \mathbf{C} \cdot \mathbf{C}^* = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} = \mathbf{I} \] Dies beweist, dass \( \mathbf{C} \) unitär ist.

Key Concepts

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

Matrix Inversion

Matrix inversion is a key concept in linear algebra. It involves finding a matrix that, when multiplied by the original matrix, yields the identity matrix. For a square matrix \(A\), its inverse \(A^{-1}\) satisfies the equation \(A \cdot A^{-1} = I\). The identity matrix \(I\) acts as a multiplicative identity in matrix multiplication.
Matrix inversion is crucial in many applications. It helps in solving linear equations, analyzing systems, and performing transformations in various fields like physics and engineering. However, not all matrices are invertible. A matrix must be square (same number of rows and columns) and have a non-zero determinant to have an inverse.
When dealing with unitary matrices, the inversion becomes straightforward. The inverse of a unitary matrix is simply its conjugate transpose, making calculations more manageable and efficient.

Conjugate Transpose

The conjugate transpose, also known as the Hermitian transpose, involves two steps: taking the transpose of a matrix and then taking the complex conjugate of each entry. For a matrix \(A\), its conjugate transpose is denoted as \(A^*\) or \(A^H\).
In mathematical notation, if \(A = [a_{ij}]\), then \(A^* = [a_{ji}^*]\), where \(a_{ji}^*\) is the complex conjugate of \(a_{ji}\).
Conjugate transposes are particularly important in the context of unitary matrices and complex matrices. A unitary matrix \(U\) satisfies the condition \(U \cdot U^* = I\), where \(I\) is the identity matrix. This property makes unitary matrices valuable in quantum mechanics and signal processing, where preserving length and angles of vectors is crucial.

Identity Matrix

The identity matrix, denoted by \(I\), is a special square matrix with ones on the main diagonal and zeros elsewhere. Its main significance lies in its property as the multiplicative identity in matrix multiplication. This means any matrix \(A\) multiplied by the identity matrix \(I\) remains unchanged: \(A \cdot I = A\).
The identity matrix is crucial in defining matrix inversion and unitary matrices. It serves as a reference point and is used to verify various properties of other matrices.

  • For a 2x2 matrix, the identity matrix is \[ I = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \]
  • For a 3x3 matrix, it is \[ I = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix} \]

The identity matrix plays a pivotal role in solving linear systems and performing matrix transformations.

Linear Algebra

Linear algebra is the branch of mathematics concerned with vectors, vector spaces, and linear transformations. It provides tools and concepts essential for understanding the structure of space and solving systems of linear equations.
Key elements of linear algebra include:

  • Vectors and vector spaces: Collections of vectors that can be scaled and added together.
  • Matrices: 2D arrays of numbers that represent linear transformations between vector spaces.
  • Eigenvalues and eigenvectors: Scalars and vectors associated with a matrix that describe its fundamental properties.
  • Determinants: Scalar values that determine whether a matrix is invertible.

Linear algebra is foundational for various disciplines, including physics, computer science, engineering, and economics. Understanding its concepts, such as unitary matrices, matrix inversion, and the identity matrix, is crucial for advancing in these fields.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Problem 8 Zeigen Sie, daB die folgenden Ma... [FREE SOLUTION] (3)

Most popular questions from this chapter

Zeigen Sie, daB die drei r?umlichen Vektoren $$ \mathbf{a}=\left(\begin{array}{l} 1 \\ 4 \\ 1 \end{array}\right), \quad \mathbf{b}=\left(\begin{array}{r} 2 \\ -1 \\ 2 \end{array}\right) \quad \text { und } \quad c=\left(\begin{array}{r} 10 \\ 4 \\ 10 \end{array}\right) $$ komplanar sind, d.h. in einer gemeinsamen Ebene liegen.Zeigen Sie die lineare Unabh?ngigkeit der folgenden Vektoren: a) \(\mathbf{a}_{1}=\left(\begin{array}{l}2 \\ 1 \\ 0\end{array}\right),\quad \mathbf{a}_{2}=\left(\begin{array}{r}-1 \\ 2 \\ 5\end{array}\right),\quad \mathbf{a}_{3}=\left(\begin{array}{r}-1 \\ 2 \\ -1\end{array}\right)\) b) \(\quad \mathbf{a}=\left(\begin{array}{r}-1 \\ 6 \\ 4\end{array}\right),\quad \mathbf{b}=\left(\begin{array}{r}1 \\ -2 \\ -2\end{array}\right),\quad \mathbf{c}=\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right)\) c) \(a=\left(\begin{array}{l}1 \\ 0 \\ 4\end{array}\right), \quadb=\left(\begin{array}{l}2 \\ 3 \\ 1\end{array}\right)\)
See all solutions

Recommended explanations on Physics Textbooks

Electricity and Magnetism

Read Explanation

Geometrical and Physical Optics

Read Explanation

Torque and Rotational Motion

Read Explanation

Linear Momentum

Read Explanation

Space Physics

Read Explanation

Measurements

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free

This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. Accept

Privacy & Cookies Policy

Privacy Overview

This website uses cookies to improve your experience while you navigate through the website. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website. These cookies will be stored in your browser only with your consent. You also have the option to opt-out of these cookies. But opting out of some of these cookies may affect your browsing experience.

Necessary

Always Enabled

Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that ensures basic functionalities and security features of the website. These cookies do not store any personal information.

Non-necessary

Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. It is mandatory to procure user consent prior to running these cookies on your website.

Problem 8 Zeigen Sie, daB die folgenden Ma... [FREE SOLUTION] (2024)

References

Top Articles
Latest Posts
Article information

Author: Aracelis Kilback

Last Updated:

Views: 5843

Rating: 4.3 / 5 (44 voted)

Reviews: 83% of readers found this page helpful

Author information

Name: Aracelis Kilback

Birthday: 1994-11-22

Address: Apt. 895 30151 Green Plain, Lake Mariela, RI 98141

Phone: +5992291857476

Job: Legal Officer

Hobby: LARPing, role-playing games, Slacklining, Reading, Inline skating, Brazilian jiu-jitsu, Dance

Introduction: My name is Aracelis Kilback, I am a nice, gentle, agreeable, joyous, attractive, combative, gifted person who loves writing and wants to share my knowledge and understanding with you.