Question

Consider the linear system \(A \mathbf{x}=\mathbf{b}\), where \(A\) is a symmetric matrix. Suppose that \(M-N\) is a splitting of \(A\), where \(M\) is symmetric positive definite and \(N\) is symmetric. Show that if \(\lambda_{\min }(M)>\rho(N)\), then the iterative scheme \(M \mathbf{x}_{k+1}=N \mathbf{x}_{k}+\mathbf{b}\) converges to \(\mathbf{x}\) for any initial guess \(\mathbf{x}_{0}\).

Short answer

Question: Prove that if the smallest eigenvalue of a symmetric positive definite matrix \(M\), \(\lambda_{\min}(M)\), is greater than the spectral radius of a symmetric matrix \(N\), \(\rho(N)\), then the iterative scheme \(M \mathbf{x}_{k+1} = N \mathbf{x}_{k} + \mathbf{b}\) converges to the exact solution \(\mathbf{x}\) for any initial guess \(\mathbf{x}_0\).

Step-by-step solution

Define variables and assumptions

Let \(A \in \mathbb{R}^{n \times n}\) be a symmetric matrix, and let \(M - N\) be a splitting of \(A\) with \(M\) symmetric positive definite and \(N\) symmetric.

Introduce iterative scheme

We can describe the iterative method for solving the linear system \(A\mathbf{x}=\mathbf{b}\) by the iterative scheme \(M\mathbf{x}_{k+1}=N\mathbf{x}_{k}+\mathbf{b}\).

Rewrite the iterative scheme in terms of errors

Let \(\mathbf{e}_k = \mathbf{x} - \mathbf{x}_k\) represent the error at iteration \(k\). Then we can rewrite the iterative scheme in terms of errors \(M\mathbf{e}_{k+1} = N\mathbf{e}_{k}\). Then, \[\mathbf{e}_{k+1} = M^{-1}N\mathbf{e}_{k} = G\mathbf{e}_{k}.\]

Find the precondition for convergence

For the iterative scheme to converge, the spectral radius of the iteration matrix \(G\) must be less than 1, i.e., \(\rho(G) < 1\).

Use Rayleigh quotient to find the minimum eigenvalue of \(M\)

Define \(\lambda_{\min}(M)\) as the smallest eigenvalue of \(M\). By using Rayleigh quotient, we can express \(\lambda_{\min}(M)\) as \[\lambda_{\min}(M) = \min_{\mathbf{y} \neq \mathbf{0}} \frac{\mathbf{y}^T M\mathbf{y}}{\mathbf{y}^T \mathbf{y}}.\]

Show \(\rho(G) < 1\) given \(\lambda_{\min}(M)>\rho(N)\)

Our goal is to show that if \(\lambda_{\min}(M)>\rho(N)\), then \(\rho(G) < 1\). Given that \(G=M^{-1}N\), we can write: \[\rho(G) = \rho(M^{-1}N) \leq \|M^{-1}N\|.\] Now, by using the property that the spectral radius \(\rho(N)\) is equal to the maximum absolute value of the eigenvalues, and using the triangle inequality, we have \(\|M^{-1}N\| \leq \|M^{-1}\|\|N\|=\frac{\|N\|}{\lambda_{\min}(M)} < 1\). Thus, if \(\lambda_{\min}(M)>\rho(N)\), we have \(\rho(M^{-1}N) < 1\), which implies \(\rho(G) < 1\).

Conclude the convergence

Since we have shown that our precondition \(\rho(G) < 1\) holds given \(\lambda_{\min}(M)>\rho(N)\), we can conclude that this iterative scheme converges to the exact solution \(\mathbf{x}\) for any initial guess \(\mathbf{x}_{0}\).

Key concepts

Symmetric Matrix

Understanding what a symmetric matrix is can greatly aid in tackling various mathematical problems, especially in the realm of linear algebra. A symmetric matrix is a square matrix, meaning it has the same number of rows and columns, and it mirrors itself across the main diagonal. In simpler terms, the matrix element at row i and column j is equal to the element at row j and column i. This can be denoted mathematically as:

• If matrix A is symmetric, then for all i and j, \( a_{ij} = a_{ji} \).

This property makes symmetric matrices considerably easier to work with, particularly when finding their eigenvalues and eigenvectors. In many applications, symmetric matrices arise naturally or are used because they simplify both theory and computation.

Positive Definite Matrix

A positive definite matrix is a special type of symmetric matrix that is crucial in ensuring certain properties, like convergence, in iterative methods. A matrix \( M \) is positive definite if it is symmetric and for any non-zero vector \( \mathbf{y} \), the quadratic form \( \mathbf{y}^T M \mathbf{y} \) is always positive:

• Mathematically, \( M \) is positive definite if \( \mathbf{y}^T M \mathbf{y} > 0 \) for all \( \mathbf{y} eq \mathbf{0} \).

Positive definite matrices are significant in optimization and machine learning since they ensure unique solutions to problems and stability in algorithms. They also guarantee that many properties of matrices and operations, like matrix inversions, are well-defined and numerically stable.

Spectral Radius

The spectral radius is a concept that helps measure how a matrix behaves in terms of its eigenvalues. It is defined as the largest absolute value of all the eigenvalues of a matrix. Knowing the spectral radius of a matrix is crucial, for instance, to determine the convergence of an iterative algorithm:

• The spectral radius \( \rho(N) \) of matrix \( N \) is given by \( \rho(N) = \max | \lambda_i | \), where \( \lambda_i \) are the eigenvalues of \( N \).

If the spectral radius of a matrix used in an iterative method is less than one, the method is guaranteed to converge, which is why this property was used in the exercise context to establish convergence criteria.

Eigenvalues

Eigenvalues are fundamental in understanding matrix transformations and characteristics in linear algebra. For a square matrix \( A \), an eigenvalue \( \lambda \) is a scalar such that there is a non-zero vector \( \mathbf{v} \), called the eigenvector, satisfying:

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

The significance of eigenvalues is vast; they describe things like:

• Matrix stability and invertibility
• The behavior of dynamical systems
• The convergence criteria of iterative methods like the ones discussed in the exercise

In symmetric matrices, all eigenvalues are real numbers, which simplifies analysis and computation. Thus, understanding eigenvalues can provide deep insights into the working of different mathematical processes and algorithms.