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Magazine Chapter 9: Problem 8
Find a unitary matrix U and a diagonal matrix \(D\) such that \(D=U^{-1} A U\) for the given matrix \(A\).\(A=\left[\begin{array}{ccc}0 & 0 & 1+2 i \\ 0 & 5 & 0 \\\ 1-2 i & 0 & 4\end{array}\right]\)
Short Answer
Expert verified
Find eigenvalues and eigenvectors of \( A \), normalize eigenvectors to form \( U \), use these to form \( D \).
Step by step solution
01
Understand Unitary and Diagonal Matrix
A unitary matrix \( U \) is a complex square matrix that satisfies \( UU^* = I \), where \( U^* \) is the conjugate transpose of \( U \) and \( I \) is the identity matrix. A diagonal matrix \( D \) has non-zero elements only along its main diagonal.
02
Find Eigenvalues of A
To find the eigenvalues of matrix \( A \), solve the characteristic equation \( \det(A - \lambda I) = 0 \). The eigenvalues of \( A \) will be the diagonal elements of \( D \).
03
Compute Eigenvectors of A
For each eigenvalue \( \lambda \), solve \((A - \lambda I)\mathbf{v} = 0 \) to find the corresponding eigenvector \( \mathbf{v} \). These vectors will become the columns of \( U \).
04
Normalize Eigenvectors
Normalize each obtained eigenvector to ensure that \( U \) is a unitary matrix. A normalized vector \( \mathbf{v} \) satisfies \( ||\mathbf{v}|| = 1 \).
05
Form Diagonal Matrix D and Unitary Matrix U
Arrange the eigenvalues along the diagonal of matrix \( D \), and use the normalized eigenvectors as columns of matrix \( U \).
06
Verify Unitary Condition
Check that \( UU^* = I \) to ensure \( U \) is indeed a unitary matrix.
07
Verify Diagonalization
Calculate \( D = U^* A U \) to ensure that \( A \) is diagonalized by \( U \) and the diagonal elements correspond to the eigenvalues of \( A \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Eigenvalues
In linear algebra, understanding eigenvalues is crucial when dealing with matrix transformations. **Eigenvalues** are special scalars associated with a square matrix that provide insightful information about the matrix's behavior.
Imagine stretching or squishing a vector with specific intensity or direction. An eigenvalue represents this scaling factor when a matrix multiplies a vector.
Imagine stretching or squishing a vector with specific intensity or direction. An eigenvalue represents this scaling factor when a matrix multiplies a vector.
- To compute an eigenvalue, solve the characteristic equation: \( \det(A - \lambda I)=0 \).
- Here, \( \lambda \) represents the eigenvalue, \( A \) is the given matrix, and \( I \) is the identity matrix of compatible size.
- The result is a polynomial equation; the roots of this polynomial are your eigenvalues.
- In some cases, these could be complex numbers, especially if the matrix entries themselves are complex.
Eigenvectors
Once you've found the eigenvalues, you can determine the **eigenvectors**. Eigenvectors, when transformed by a matrix, scale by their corresponding eigenvalues.
These vectors don't change direction, only their magnitude changes.
These vectors don't change direction, only their magnitude changes.
- To find eigenvectors, solve \((A - \lambda I)\mathbf{v} = 0 \).
- \( \mathbf{v} \) represents the eigenvector, \( \lambda \) is an eigenvalue, and \( I \) is the identity matrix.
- Each eigenvalue typically has one or more associated eigenvectors.
Diagonalization
**Diagonalization** is the process of finding a diagonal matrix \( D \) from a given matrix \( A \) using its eigenvalues and eigenvectors.
This transforms \( A \) into a simpler form which is easy to work with for complex operations like powers or exponentials.
The process follows these steps:
This transforms \( A \) into a simpler form which is easy to work with for complex operations like powers or exponentials.
The process follows these steps:
- Form the unitary matrix \( U \) using its columns as the normalized eigenvectors of \( A \).
- The matrix \( D \) will have the eigenvalues of \( A \) placed along its diagonal.
- Verify that \( D = U^* A U \), where \( U^* \) is the conjugate transpose of \( U \).
Normalization
The process of **normalization** ensures that all eigenvectors have a unit length. This is essential to forming a unitary matrix.
If eigenvectors are not normalized, they can stretch or compress the transformation unpredictably.
If eigenvectors are not normalized, they can stretch or compress the transformation unpredictably.
- A normalized vector \( \mathbf{v} \) satisfies \(||\mathbf{v}|| = 1\), meaning the length or magnitude of \( \mathbf{v} \) is 1.
- To normalize a vector, divide it by its own magnitude \(||\mathbf{v}||\).
- Normalization is critical in constructing matrices that maintain orthogonality, ensuring the unitary condition \( UU^* = I \).
