Question

(a) Show that if the square matrix \(A\) is strictly diagonally dominant, then the Jacobi relaxation yields an iteration matrix that satisfies $$ \|T\|_{\infty}<1 $$ (b) Show that if \(A\) is a \(2 \times 2\) symmetric positive definite matrix, the Jacobi scheme converges for any initial guess.

Short answer

Question: Prove that the Jacobi relaxation method (a) converges when applied to a strictly diagonally dominant matrix and (b) converges for any initial guess when the matrix A is a symmetric positive definite 2x2 matrix. Answer: (a) For a strictly diagonally dominant matrix, the infinity-norm of the iteration matrix T is less than 1, which ensures the convergence of the Jacobi relaxation method. (b) For a 2x2 symmetric positive definite matrix, the eigenvalues of the iteration matrix T are found to be purely imaginary with a magnitude less than 1, which implies that the spectral radius of T is less than 1. This guarantees the convergence of the Jacobi scheme for any initial guess when the matrix A is a symmetric positive definite 2x2 matrix.

Step-by-step solution

Recall the Jacobi relaxation method and its iteration matrix

The Jacobi relaxation method can be applied to solve a linear system of the form \(Ax=b\), with A being an \(n \times n\) matrix and x, b being column vectors of length n. The method is based on the idea of splitting the matrix A into a diagonal matrix D and a remainder matrix R, such that A = D + R. The iteration formula for the Jacobi method can be written as follows: $$ x^{(k+1)} = Dx^{-1}(b - Rx^{(k)}) $$ This equation can be rewritten in the form of the iteration matrix and the constant vector: $$ x^{(k+1)} = Tx^{(k)}+c $$ Here, the iteration matrix T is given by $$ T = -D^{-1}R $$ The convergence of the Jacobi method is determined by the properties of the iteration matrix T.

Analyze strictly diagonally dominant matrices

A square matrix A is called strictly diagonally dominant if for every row i, the absolute value of the diagonal element is strictly greater than the sum of the absolute values of the off-diagonal elements in the same row: $$ |a_{ii}| > \sum_{j \neq i} |a_{ij}| $$ For all \(i=1,2,...,n\).

Show that the infinity-norm of T is less than 1 for strictly diagonally dominant A

Now we need to show that if A is a strictly diagonally dominant matrix, the infinity-norm of the iteration matrix T is less than 1. First, recall the definition of the infinity-norm of a matrix: $$ \|T\|_\infty = \max_{1\leq i\leq n}\sum_{j=1}^n|t_{ij}| $$ Substitute the formula for T: $$ \|T\|_\infty = \max_{1\leq i\leq n}\sum_{j=1}^n\left|\frac{-a_{ij}}{a_{ii}}\right|, \ j \neq i $$ Now, observe the sum for i given some row i: $$ \sum_{j=1}^n\left|\frac{-a_{ij}}{a_{ii}}\right| = \frac{1}{|a_{ii}|} \sum_{j\neq i} |a_{ij}| $$ Since A is strictly diagonally dominant, we know that \(|a_{ii}| > \sum_{j\neq i} |a_{ij}|\). Therefore, the sum \(\sum_{j\neq i} |a_{ij}| < |a_{ii}|\), and since \(a_{ii} \neq 0\), the fraction \(\frac{\sum_{j\neq i} |a_{ij}|}{|a_{ii}|} < 1\). Hence, for any row i, the sum of the magnitudes of the off-diagonal elements divided by the magnitude of the diagonal element is less than 1. Now, let's find the maximum of these fractions across all rows i: $$ \|T\|_\infty = \max_{1\leq i\leq n} \left(\frac{1}{|a_{ii}|} \sum_{j\neq i} |a_{ij}|\right) < 1 $$ This shows that the infinity-norm of the iteration matrix T is less than 1 if the matrix A is strictly diagonally dominant.

Show convergence of the Jacobi scheme for a 2x2 symmetric positive definite A

Now consider a 2x2 symmetric positive definite matrix A: $$ A = \begin{bmatrix} a & b \\ b & c \end{bmatrix} $$ A is positive definite so \(a>0, c>0\), and the determinant of A: \(ac - b^2>0\). Also, note that b is strictly less than the maximum of a and c, since the determinant of A has to be greater than 0. Let the j-th row sum in absolute value be \(S_j = \sum_{i\neq j} |a_{ij}| = |a_{ij}|\) for j=1,2. Now we check the eigenvalues of T. The matrix T has the form: $$ T = \begin{bmatrix} 0 & -\frac{b}{a} \\ -\frac{b}{c} & 0 \end{bmatrix} $$ The eigenvalues of T are the solutions of the following characteristic equation: $$ det(T-\lambda I) = \lambda^2 + \frac{b^2}{ac} = 0 $$ Using the quadratic formula, we can find the eigenvalues as $$ \lambda = \pm\sqrt{-\frac{b^2}{ac}} $$ Both eigenvalues are purely imaginary, and have a magnitude of $$ |\lambda| = \sqrt{\frac{b^2}{ac}} < \sqrt{1} = 1 $$ Since the spectral radius \(\rho(T) = \max\{|\lambda_{1}|, |\lambda_{2}|\} < 1\), the Jacobi scheme converges for any initial guess when A is a 2x2 symmetric positive definite matrix.

Key concepts

Diagonally Dominant Matrix

A matrix is considered diagonally dominant when each row of the matrix has a strong emphasis on its diagonal element compared to the sum of its off-diagonal elements. More formally, a matrix \(A = [a_{ij}]\) is strictly diagonally dominant if for each row \(i\), the following inequality holds: \[ |a_{ii}| > \sum_{j eq i} |a_{ij}| \]This condition ensures that the influence of the diagonal element in each row is more significant than any possible combination of the other elements in that row. Why does this matter? For many iterative methods, such as the Jacobi method, having a strictly diagonally dominant matrix guarantees convergence. This means that as you apply the iterative method starting from an initial guess, you will converge to the correct solution. Without this dominance, convergence is not assured, and the algorithm may fail to find the solution.

Convergence of Iterative Methods

Iterative methods are designed to solve linear systems using successive approximations rather than direct computation. The Jacobi Iteration Method is a popular technique, particularly appealing for its simplicity and parallelization capability. Nonetheless, these methods do not universally converge.Convergence is crucial as it indicates whether the iterative methods will actually yield a solution as iterations progress. Specifically, for methods like Jacobi, convergence is often checked using the iteration matrix, \( T = -D^{-1}R \), where \( D \) is the diagonal part of \( A \) and \( R \) contains the remaining elements. An iterative method converges if the spectral radius (the largest absolute value of the eigenvalues) of \( T \) is less than 1. In practical terms:

• Choose a method and initial guess.
• Establish conditions like diagonal dominance or positive definiteness.
• Iterate until convergence is achieved.

With these checks, one can predict and ensure that a given iterative scheme will converge.

Matrix Norms

Matrix norms are tools to measure the size or length of a matrix, and they come in various forms: the 1-norm, the 2-norm (also called the Frobenius norm in matrices), and the infinity norm are a few common ones. The infinity norm of a matrix, \( \|A\|_\infty \), is defined as the maximum absolute row sum of the matrix. For a matrix \( A \),\[\|A\|_\infty = \max_{1\leq i\leq n}\sum_{j=1}^n|a_{ij}|\]This norm provides a way to analyze matrices, especially when checking for convergence of iterative techniques. In the case of the Jacobi method, checking if \( \|T\|_\infty < 1 \) helps confirm that the iteration matrix \( T \) will converge the method.In summary, understanding these norms helps gauge the effect of each matrix operation and determines the numerical stability and convergence of algorithms.

Symmetric Positive Definite Matrices

A matrix is symmetric if it equals its transpose, and it is positive definite if all its eigenvalues are positive. These matrices play a vital role in numerical analysis, as they possess properties that facilitate efficient solving of linear systems.For a 2x2 matrix \( A \) given by:\[A = \begin{bmatrix} a & b \ b & c \end{bmatrix}\]being symmetric positive definite means:

• \( a > 0 \) and \( c > 0 \)
• The determinant, \( ac - b^2 > 0 \)

These properties ensure that the Jacobi method will converge for any initial guess. The practical implication is that if a matrix possesses these characteristics, one can confidently choose the Jacobi method for solving the system, confident in rapidly reaching an accurate solution.