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

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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


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.


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

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


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.


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

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


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.


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

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


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.

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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) \(a=\left(\begin{array}{l}1 \\ 0 \\ 4\end{array}\right), \quadb=\left(\begin{array}{l}2 \\ 3 \\ 1\end{array}\right)\)
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Problem 8 Zeigen Sie, daB die folgenden Ma... [FREE SOLUTION] (2024)


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