Question

In each case, find a unitary matrix \(U\) such that \(U^{H} A U\) is diagonal. a. \(A=\left[\begin{array}{rr}1 & i \\ -i & 1\end{array}\right]\) b. \(A=\left[\begin{array}{cc}4 & 3-i \\ 3+i & 1\end{array}\right]\) c. \(A=\left[\begin{array}{rr}a & b \\ -b & a\end{array}\right] ; a, b,\) real d. \(A=\left[\begin{array}{cc}2 & 1+i \\ 1-i & 3\end{array}\right]\) e. \(A=\left[\begin{array}{ccc}1 & 0 & 1+i \\ 0 & 2 & 0 \\ 1-i & 0 & 0\end{array}\right]\) f. \(A=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 1+i \\ 0 & 1-i & 2\end{array}\right]\)

Short answer

Find eigenvalues and eigenvectors, then construct \( U \) with normalized eigenvectors.

Step-by-step solution

Identify Hermitian Matrix

A matrix is Hermitian if it is equal to its own conjugate transpose. In each case, verify Hermiticity by checking if \( A = A^H \).

Find Eigenvalues

Compute the eigenvalues of each matrix \( A \). For a 2x2 matrix, use the characteristic polynomial \( \text{det}(A - \lambda I) = 0 \). Solve this polynomial to find eigenvalues \( \lambda_1 \) and \( \lambda_2 \).

Find Eigenvectors

For each eigenvalue \( \lambda \), solve the system \( (A - \lambda I) \mathbf{v} = \mathbf{0} \) to find the corresponding eigenvector \( \mathbf{v} \). Normalize these eigenvectors to ensure they form an orthonormal set, since \( U \) is unitary and its columns must be orthonormal.

Formulate Unitary Matrix

Construct the unitary matrix \( U \), whose columns are the normalized eigenvectors found in Step 3. Ensure that \( U \) is unitary, i.e., \( U^H U = I \), where \( I \) is the identity matrix.

Confirm Diagonalization

Verify that \( U^H A U \) results in a diagonal matrix. This matrix has the eigenvalues of \( A \) on its diagonal, confirming the diagonalization.

Solve for Each Matrix

Apply Steps 1 through 5 individually to each given matrix from parts (a) through (f), ensuring to check calculations for eigenvalues, eigenvectors, and construct \( U \) appropriately for each case.

Key concepts

Hermitian Matrix

A Hermitian matrix is a special type of square matrix that is equal to its own conjugate transpose. This property means that a Hermitian matrix has complex entries where the element at the ith row and jth column is the complex conjugate of the element at the jth row and ith column. The formula for this condition is given as:

• For a matrix \( A \), the matrix is Hermitian if \( A = A^H \), where \( A^H \) denotes the conjugate transpose of \( A \).

Hermitian matrices are important because they have real eigenvalues and their eigenvectors can be chosen to be orthogonal. This is a crucial property in many fields such as quantum mechanics and vibration analysis.
When dealing with a Hermitian matrix, the entries on the main diagonal are always real numbers. Checking if a matrix is Hermitian is the first step in solving problems related to matrix diagonalization.

Eigenvalues

Eigenvalues are special numbers associated with a matrix that provide insight into the matrix's fundamental properties. For any square matrix \( A \), an eigenvalue \( \lambda \) satisfies the equation \( A \mathbf{v} = \lambda \mathbf{v} \), where \( \mathbf{v} \) is the eigenvector.

• To find the eigenvalues of a matrix, solve the characteristic equation: \( \text{det}(A - \lambda I) = 0 \).

This equation is derived from setting the determinant of \( A - \lambda I \) equal to zero, ensuring that there is a non-trivial solution for the eigenvector \( \mathbf{v} \).
Eigenvalues can reveal a lot about a matrix's behavior, such as stability, behavior under various transformations, and response to different stimuli. In the case of Hermitian matrices, all eigenvalues are real, making them particularly convenient for real-world applications like vibration modes and energy levels in physics.

Eigenvectors

Eigenvectors are non-zero vectors that change at most by a scalar factor when a linear transformation is applied. For a given matrix \( A \), if \( \mathbf{v} \) is an eigenvector corresponding to the eigenvalue \( \lambda \), then:

• \( A \mathbf{v} = \lambda \mathbf{v} \).

This means applying the transformation represented by matrix \( A \) to \( \mathbf{v} \) stretches or shrinks \( \mathbf{v} \) by the factor \( \lambda \).
Finding an eigenvector involves solving the equation \( (A - \lambda I) \mathbf{v} = \mathbf{0} \). The solution provides the direction along which the matrix transformation has uniform stretching.
For Hermitian matrices, eigenvectors corresponding to different eigenvalues are orthogonal. This orthogonality is critical when forming the unitary matrix \( U \) used in diagonalizing Hermitian matrices. Normalizing these orthogonal eigenvectors ensures they are suitable for constructing a unitary matrix.

Diagonalization

Diagonalization is the process of finding a basis under which a matrix is represented by a diagonal matrix. This is done by transforming the matrix with a unitary matrix \( U \). The transformation is expressed as \( U^H A U = D \), where \( D \) is a diagonal matrix.
Diagonalization is possible for square matrices that can be expressed in terms of their eigenvalues and eigenvectors. It simplifies many matrix operations, such as exponentiation, because calculations with diagonal matrices are straightforward.

• For Hermitian matrices, diagonalization guarantees all diagonal elements are real eigenvalues of the original matrix.

To achieve diagonalization:

• Construct the unitary matrix \( U \) using the normalized eigenvectors of \( A \).
• Ensure \( U^H U = I \), confirming \( U \) is unitary.
• Verify that \( U^H A U \) is diagonal, with eigenvalues on the diagonal.

This process of diagonalization is extensively used in numerical analysis, quantum mechanics, and system theory to make complex linear algebra problems more manageable.