Question

If \(A\) is symmetric, show that each matrix \(A_{k}\) in the QR-algorithm is also symmetric. Deduce that they converge to a diagonal matrix.

Short answer

Each matrix in the QR-algorithm remains symmetric, and they converge to a diagonal matrix.

Step-by-step solution

Understanding Symmetric Matrices

A matrix is symmetric if it is equal to its transpose, i.e., for a symmetric matrix \(A\), we have \(A = A^T\). This property will be used to show that each matrix in the QR-algorithm is symmetric.

Introduction to the QR-algorithm

The QR-algorithm is an iterative method used for finding the eigenvalues of a matrix. Starting with a symmetric matrix \(A\), we perform a QR decomposition to write \(A = QR\), where \(Q\) is an orthogonal matrix and \(R\) is an upper triangular matrix. We then form a new matrix \(A_1 = RQ\).

Prove Symmetry in Subsequent Matrices

Assuming \(A_k\) is symmetric, we perform a QR decomposition: \(A_k = Q_k R_k\). The next matrix in the algorithm is \(A_{k+1} = R_k Q_k\). We need to show that \(A_{k+1}^T = A_{k+1}\).

Utilize Orthogonality of Q

Since \(Q_k\) is orthogonal, \(Q_k^T = Q_k^{-1}\). We have \(A_{k+1} = R_k Q_k\), and its transpose is \(A_{k+1}^T = (R_k Q_k)^T = Q_k^T R_k^T = Q_k^{-1} R_k^T\). Using \(A_k = Q_k R_k\) and \(A_k^T = A_k = R_k^T Q_k^T\), we recognize that symmetry preserves this form, so \(A_{k+1}\) is symmetric.

Conclude the Matrices Converge

As the matrices \(A_k\) remain symmetric, the QR algorithm's sequence of transformations will eventually lead the matrices to a diagonal form, as the iteratively constructed sequences preserve eigenvalue and similar structure. The result of continuously applying QR factorizations on symmetric matrices is diagonalization.

Key concepts

Symmetric Matrices

A symmetric matrix is an essential concept in linear algebra, where the matrix equals its own transpose. In other words, for a matrix \( A \), it is symmetric if \( A = A^T \). To easily check for symmetry, look at the elements across the diagonal. If \( a_{ij} = a_{ji} \) for all \( i \) and \( j \), then the matrix is symmetric.

Symmetric matrices possess special properties which make them very useful in mathematics and engineering. For one, they ensure that their eigenvalues are always real numbers, not complex numbers. Additionally, symmetric matrices can be diagonalized using an orthogonal matrix. The symmetry leads to simplified calculations for algorithms designed to find eigenvalues, like the QR algorithm.

Understanding symmetric matrices is often the first step when dealing with iterative matrix algorithms, such as the QR algorithm. In cases where the algorithm starts with a symmetric matrix, all subsequent matrices produced by the algorithm also maintain symmetry due to mathematical consistency and properties of the operations involved.

Eigenvalues

Eigenvalues are intrinsic values associated with a matrix that provide insight into the matrix's characteristics. If \( A \) is a square matrix, an eigenvalue \( \lambda \) meets the condition \( A\mathbf{v} = \lambda\mathbf{v} \), where \( \mathbf{v} \) is a non-zero vector known as an eigenvector. These values allow one to understand various properties like stability and oscillation modes in physical systems.

In symmetric matrices, as mentioned earlier, eigenvalues are always real. This is a crucial feature because it simplifies analysis and computations significantly. When it comes to the QR algorithm, it leverages eigenvalues by iteratively breaking down and reconstructing matrices to eventually diagonalize them, which reveals the eigenvalues along the diagonal.

The QR algorithm, specifically, starts with a symmetric matrix and through a series of transformations tends to preserve the eigenvalues while transforming each successive matrix in the algorithm closer to a diagonal form. This process not only highlights the eigenvalues but also confirms their stability across iterations.

Orthogonal Matrices

Orthogonal matrices play a pivotal role in linear algebra and matrix analysis, particularly in algorithms such as the QR algorithm. An orthogonal matrix \( Q \) satisfies the condition \( Q^T = Q^{-1} \). In simpler terms, its transpose is the same as its inverse. This property ensures that multiplying an orthogonal matrix by its transpose results in an identity matrix \( I \).

Orthogonality preserves both angles and lengths, an essential feature when applied within algorithms that require stability over iterative processes. The QR algorithm uses orthogonal matrices to maintain symmetry and preserve eigenvalues throughout its iterations. Specifically, each step of the QR decomposition involves forming an orthogonal matrix \( Q \) and an upper triangular matrix \( R \), such that \( A = QR \) and then re-forming a new matrix \( A_1 = RQ \).

The orthogonal property's stability ensures that each iteration in the QR algorithm results in matrices that remain symmetric if they initially start symmetric. This maintained symmetry enhances the progression towards diagonal matrix forms, which are ultimately the target of the algorithm for efficient and accurate eigenvalue extraction.