Question

Show that an arbitrary matrix A can be "diagonalized" as \(\mathrm{D}=\) UAV, where \(U\) is unitary and \(D\) is a real diagonal matrix with only nonnegative eigenvalues. Hint: There exists a unitary matrix that diagonalizes \(\mathrm{AA}^{\dagger}\).

Short answer

\(A\) can be diagonalized as \(D = UAV\), where \(D\) is a diagonal matrix with nonnegative eigenvalues on the main diagonal, and \(U\) and \(V\) are unitary matrices; \(U\) is formed from the eigenvectors of \(AA^{\dagger}\).

Step-by-step solution

Eigenvalues and Eigenvectors

Do the eigenvalue equation on \(AA^{\dagger}\). The eigenvalues are real and nonnegative. For each eigenvalue, find a corresponding eigenvector.

Formation of Unitary matrix U

Create a matrix U with the normalized eigenvectors found in Step 1 as its columns.

Confirming \(U\) is Unitary

Show that the formed matrix U is a unitary matrix. You do this by showing that \(UU^{\dagger}=I\). Where I is the identity matrix and U is the formed matrix from step 2.

Diagonalization

Diagonalize matrix A. Prove that \(D = UAU^{\dagger}\), where D is a diagonal matrix with the eigenvalues in the diagonal.

Final Proof

Substitute \(AA^{\dagger}\) for D to prove that \(D = UAV\).

Key concepts

Unitary Matrix

A unitary matrix is a square matrix that has a very special property: when it is multiplied by its conjugate transpose, the result is the identity matrix. In simpler terms, a unitary matrix represents a structure-preserving transformation in complex vector spaces—that is, one that keeps the 'length' of complex vectors unchanged after transformation. This is akin to rotating or reflecting objects without changing their size in real space.

The conjugate transpose of a matrix, often denoted as \(A^\dagger\), is achieved by taking the transpose of the matrix—flipping it over its diagonal—and then taking the complex conjugate of each element. The complex conjugate involves changing the sign of the imaginary part of a complex number (for instance, the complex conjugate of \(3+4i\) is \(3-4i\)).

In our case, confirming that a matrix \(U\) is unitary is an essential step to ensure the preservation of vector norms through the transformation. Remembering that for every unitary matrix, \(UU^\dagger = I\), where \(I\) is the identity matrix, is critical in both understanding and proving unitary properties during diagonalization or other matrix operations.

Eigenvalues and Eigenvectors

Diving into the heart of matrix operations, we encounter the crucial concepts of eigenvalues and eigenvectors. These form the foundation of what it means to 'diagonalize' a matrix. When we apply a matrix to one of its eigenvectors, the vector may stretch or shrink, but it doesn't change its direction. This scalar change in length is termed as an eigenvalue.

The eigenvalue equation, symbolically \(A\vec{x} = \lambda\vec{x}\), defines the relationship between a matrix \(A\), an eigenvector \(\vec{x}\), and its corresponding eigenvalue \(\lambda\). In essence, eigenvectors are vectors that remain parallel to themselves even after a linear transformation by \(A\) is applied, and eigenvalues indicate how these eigenvectors are scaled.

In connection to the given exercise, \(AA^\dagger\) ensures that the eigenvalues are indeed real and nonnegative due to the nature of conjugate transpose operations leading to a Hermitian matrix—this special type of matrix always has such eigenvalues. Each eigenvalue corresponds to a specific eigenvector, and when these eigenvectors are correctly normalized and combined, they form the columns of the unitary matrix \(U\) which is used in further diagonalization steps.

Diagonal Matrix

A diagonal matrix is a type of matrix that is very intuitive to comprehend. In this matrix, only the elements on the main diagonal can be non-zero, while all off-diagonal elements are zero. Imagine a grid where only a straight line from the top left corner to the bottom right corner is marked, and every other space is blank—that's essentially a diagonal matrix. Mathematically, it can be represented as \(D = \textrm{diag} (d_1, d_2, ..., d_n)\), where \(d_1, d_2, ..., d_n\) are the elements on the diagonal.

The beauty of a diagonal matrix lies in its simplicity. Calculating powers of the matrix, finding its determinant, or multiplying it with another matrix become significantly easier tasks. In the exercise, diagonalizing the matrix \(A\) is a crucial step to simplify complex matrix operations. This diagonal matrix consists of the eigenvalues that we've found as its diagonal entries. Since the eigenvalues are properties of \(AA^\dagger\), this ensures that the diagonalization will result in real and nonnegative diagonal elements in \(D\). The representation \(D = UAU^\dagger\) encapsulates the transformation process, where \(U\) is our unitary matrix from the previous steps.