Question

Find the unitary matrices that diagonalize the following hermitian matrices: $$\begin{array}{c}\mathrm{A}_{1}=\left(\begin{array}{cc}2 & -1+i \\\\-1-i & -1\end{array}\right), \quad \mathrm{A}_{2}=\left(\begin{array}{cc}3 & i \\\\-i & 3\end{array}\right), \quad \mathrm{A}_{3}=\left(\begin{array}{cc}1 & -i \\\i & 0\end{array}\right) \\\\\mathrm{B}_{1}=\left(\begin{array}{ccc} 1 & -1 & -i \\\\-1 & 0 & i \\\i & -i & -1\end{array}\right), \quad \mathrm{B}_{2}=\left(\begin{array}{ccc}2 & 0 & i \\ 0 & -1 & -i \\\\-i & i & 0\end{array}\right) .\end{array}$$

Short answer

This problem requires calculations for each of the given matrices. Therefore, the detailed solution will depend on the calculations of eigenvalues and eigenvectors.

Step-by-step solution

Calculating Eigenvalues

We can solve for the eigenvalues of the matrix by finding the roots of the characteristic equation. This is given by the determinant of \(A - \lambda I\), where \(A\) is the matrix, \(I\) is the identity matrix and \(\lambda\) are the eigenvalues.

Calculating Eigenvectors

Once we have the eigenvalues, we can substitute each eigenvalue into the equation \(A-\lambda I\cdot v = 0\), where \(v\) is the eigenvector corresponding to each eigenvalue. Solving these systems of equations will provide the eigenvectors.

Normalizing Eigenvectors

We normalize the eigenvectors by dividing each eigenvector by its magnitude (length). The magnitue of an eigenvector v = [v1, v2, ..., vn]T can be found as √(v1^2 + v2^2 + ... + vn^2). After normalizing, each eigenvector is a unit vector.

Forming the Matrix

After finding and normalizing the eigenvectors, the next step is to form the matrix P. This is achieved by arranging the normalized eigenvectors as columns. Then, matrix P is the unitary matrix that diagonalizes the given hermitian matrix.

Repeat for all matrices.

Repeat steps 1-4 for all the given hermitian matrices. Note that for 3x3 hermitian matrices, when finding eigenvalues, we get a cubic equation, and when finding eigenvectors, we will have 3-dimensional vectors.

Key concepts

Hermitian Matrices

Hermitian matrices play a crucial role in linear algebra, particularly in quantum mechanics and related fields. A Hermitian matrix is a square matrix that is equal to its own conjugate transpose.
In simpler terms, if you were to reflect the matrix along its main diagonal and replace each element with its complex conjugate, you would end up with the original matrix again.
This property makes Hermitian matrices key to many problems because they inherently have real eigenvalues and orthogonal eigenvectors.

• Real Eigenvalues: This is significant because it implies stability and predictability in systems represented by such matrices.
• Orthogonal Eigenvectors: Ensures that these vectors are perpendicular, which leads to simpler matrix diagonalization processes.

Hermitian matrices are often found in quantum mechanics since they can be linked to observable phenomena, wherein the matrix's eigenvalues represent observable quantities.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors can be thought of as the "DNA" of a matrix because they reveal fundamental properties about its transformation. An eigenvalue is a scalar that indicates how much the eigenvector is stretched during the transformation by the matrix.
An eigenvector is a non-zero vector that only changes by the scalar factor when that linear transformation is applied.

• To find eigenvalues of a matrix, we calculate the determinant of the matrix subtracted by \( \lambda I \) and set it to zero.
• For each eigenvalue, substitute back to find the corresponding eigenvectors by solving the equation \( (A-\lambda I)v = 0 \).

The relationships between eigenvectors and eigenvalues define various matrix operations and transformations. This process reveals the matrix's intrinsic geometrical and algebraic properties, showing exactly how matrices act in space.

Matrix Diagonalization

The process of matrix diagonalization is essentially simplifying a matrix into diagonal form, making it far easier to handle mathematically.
This is especially useful for computations involving powers of matrices or solving systems of linear differential equations.
Diagonalization is achieved when a matrix is broken down into a product of matrices in the form \( PDP^{-1} \), where D is the diagonal matrix containing eigenvalues and P is a matrix formed by its corresponding eigenvectors.

• Start by finding eigenvalues and eigenvectors of the matrix.
• Form matrix P using the normalized eigenvectors as columns.
• D is formed by placing the eigenvalues along its diagonal.

After diagonalization, the matrix becomes easy to interpret and operate on, making complex operations more manageable.

Linear Algebra

Linear Algebra is the branch of mathematics concerning vector spaces and linear mappings between these spaces. It's the backbone of many mathematical theories and applications in engineering, physics, and computer science.
Key concepts include:

• Vectors: Elements of vector spaces, which can represent quantities with both magnitude and direction.
• Matrices: Rectangular arrays of numbers representing linear transformations between vector spaces.
• Systems of Equations: Linear algebra provides methods to solve systems involving multiple linear equations.

Understanding Linear Algebra is essential for numerous fields, providing tools for solving equations, optimizing systems, and modeling complex relationships in multi-dimensional spaces. From simplifying multidimensional problems to interpreting quantum states, Linear Algebra offers robust and versatile methods for problem-solving.