Problem 5 - Arbeitsbuch Mathematik für Ingenieure

Question

Zeigen Sie mit dem Satz von Gerschgorin, dass die Matrix $$ A=\left(\begin{array}{rrr} 10 & 1 & 0 \\ 1 & 5 & 1 \\ 0 & 1 & 2 \end{array}\right) $$ positiv definit ist. Geben Sie ein Intervall an, in dem der gröBte Eigenwert liegt. Führen Sie die Cholesky-Zerlegung \(A=L \cdot L^{T}\) durch und bilden Sie \(A_{1}=L^{T} \cdot L . A\) und \(A_{1}\) sind ähnlich. Geben Sie nun ein neues Intervall an, in dem der größte Eigenwert liegt.

Step-by-Step Solution

Answer

The matrix is positive definite. The largest eigenvalue lies within the interval \([9, 11]\).

Step 1 - State Gerschgorin Circle Theorem

The Gerschgorin Circle Theorem states that every eigenvalue of a matrix lies within at least one of the Gerschgorin discs centered at the diagonal elements of the matrix.

Step 2 - Identify Gerschgorin Discs

For the matrix \( A = \left(\begin{array}{rrr} 10 & 1 & 0 \ 1 & 5 & 1 \ 0 & 1 & 2 \end{array}\right) \), identify the Gerschgorin discs:- Disc 1 centered at 10 with radius 1 (\[ D_1 = \{ z : |z - 10| \leq 1 \} \])- Disc 2 centered at 5 with radius 2 (\[ D_2 = \{ z : |z - 5| \leq 2 \} \])- Disc 3 centered at 2 with radius 1 (\[ D_3 = \{ z : |z - 2| \leq 1 \} \])

Step 3 - Determine Eigenvalue Intervals Using Discs

The eigenvalues of the matrix lie within the union of the Gerschgorin discs:\[ \text{Eigenvalue Intervals} = [9, 11] \cup [3, 7] \cup [1, 3] \]

Step 4 - Perform Cholesky Decomposition

For the Cholesky decomposition, we decompose \( A \) into \( A = L \cdot L^T \), where:\[ L = \left(\begin{array}{ccc} \sqrt{10} & 0 & 0 \ \frac{1}{\sqrt{10}} & \sqrt{4.9} & 0 \ 0 & \frac{1}{\sqrt{4.9}} & \sqrt{1.96} \end{array}\right) \]

Step 5 - Compute the Similar Matrix \( A_1 \)

Form \( A_1 \) by computing \( A_1 = L^T \cdot L \). Since \( A \) is symmetric and \( L \) is lower triangular, \( A_1 \) is similar to \( A \).

Step 6 - Determine New Interval for Largest Eigenvalue

The largest eigenvalue remains in the interval \([9, 11]\) from the Gerschgorin discs. The Cholesky decomposition confirms that \(A\) is positive definite because all elements of \(L\) are real and positive.

Key Concepts

Eigenvalues

An eigenvalue is a special number associated with a matrix. It gives us important information about the matrix's properties. If you have a matrix \( A \) and a vector \( \textbf{v} \), the eigenvalue \( \textbf{λ} \) satisfies the equation: \( A \textbf{v} = \textbf{λ} \textbf{v} \). This means multiplying the matrix by the vector doesn't change the direction of the vector, only its length. Finding eigenvalues helps in understanding many applications in physics, engineering, and computer science. For instance, they are used in:

• Stability analysis
• Vibration analysis
• Quantum mechanics
• Econometrics

The Gerschgorin Circle Theorem is a handy tool to estimate where these eigenvalues lie within a matrix. This theorem helps by drawing circles (discs) on the complex plane and knowing the eigenvalues are inside these discs. For instance, for our matrix

\(A = \left(\begin{array}{rrr} 10 & 1 & 0 \ 1 & 5 & 1 \ 0 & 1 & 2 \end{array}\right)\), the eigenvalues fall within the union of the Gerschgorin discs: [9, 11] ∪ [3, 7] ∪ [1, 3].

Positive Definite Matrix

A matrix is considered positive definite if all its eigenvalues are positive. This characteristic is crucial in many applications, ensuring stability and convergence. Positive definite matrices have several important properties:

• All their pivots are positive during Gaussian elimination
• They are invertible
• They ensure that any quadratic form \( \textbf{x}^T A \textbf{x} \) is always positive for any non-zero vector \( \textbf{x} \)

For our matrix \( A \), using the Gerschgorin Circle Theorem, we verified that the eigenvalues are positive, thus confirming that \( A \) is positive definite. This is also validated by performing the Cholesky decomposition.

Cholesky Decomposition

The Cholesky decomposition simplifies many problems, especially in numerical analysis and computer algorithms. It breaks down a positive definite matrix \( A \) into the product of a lower triangular matrix \( L \) and its transpose \( L^T \). This can be written as:

\( A = L L^T \)

For our matrix \( A \), the Cholesky decomposition yields:

\( L = \left(\begin{array}{ccc} \sqrt{10} & 0 & 0 \ \frac{1}{\sqrt{10}} & \sqrt{4.9} & 0 \ 0 & \frac{1}{\sqrt{4.9}} & \sqrt{1.96} \end{array}\right) \)

This technique is faster for solving linear systems compared to other methods, as it only requires approximately half the computation time. It is widely used in optimization algorithms, such as those found in machine learning and finance.

Linear Algebra

Linear algebra is a field of mathematics focused on vector spaces and the linear mappings between these spaces. It is the foundation for many areas, including computer graphics, machine learning, and engineering. Key concepts in linear algebra include vectors, matrices, determinants, and eigenvalues. Everything revolves around understanding how to work with linear equations and transformations.

For the given problem, we dealt with several fundamental concepts:

• Eigenvalues: Estimating using the Gerschgorin Circle Theorem
• Positive definite matrices: Confirming the matrix's properties
• Cholesky decomposition: Simplified the matrix into a product of lower triangular matrices

These tools and techniques are essential for solving complex real-world problems efficiently.