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Chapter 9: Problem 5

Bracket every eigenvalue of $$ A=\left[\begin{array}{rrrr} 2 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 1 \end{array}\right] $$

Short Answer

Expert verified
Solve characteristic equation and solve all I in it. Eigenvalues are 0, 1

Step by step solution

01

- Write Characteristic Polynomial

Begin by finding the characteristic polynomial of matrix \(A\). The characteristic polynomial is found using the determinant equation: \( \text{det}(A - \text{I} \times \text{I}) = 0 \). For matrix \(A\), this becomes: \( \text{det} \begin{bmatrix} 2-\text{I} & -1 & 0 & 0 \ -1 & 2-\text{I} & -1 & 0 \ 0 & -1 & 2-\text{I} & -1 \ 0 & 0 & -1 & 1-\text{I} \ \right] = 0 \).
02

- Find Determinant

Use cofactor expansion or row reduction to find this determinant. Start with the first row: \( (2 - \text{I}) \text{det} \begin{bmatrix} 2-\text{I} & -1 & 0 \ -1 & 2-\text{I} & -1 \ 0 & -1 & 1-\text{I} \ \right] - (-1) \text{det} \begin{bmatrix} -1 & 0 & 0 \ 2-\text{I} & -1 & 0 \ -1 & 2-\text{I} & -1 \ \right] = 0 \).
03

- Simplify Determinant of 3x3 Matrices

Solve each 3x3 determinant from Step 2: \( \text{det} \begin{bmatrix} 2-\text{I} & -1 & 0 \ -1 & 2-\text{I} & -1 \ 0 & -1 & 1-\text{I} \ \right] =(2-\text{I})[(2-\text{I})(1-\text{I}) -(-1)(-1)] - (-1)[-1(1-\text{I})] \).
04

- Solve Determinant of 3x3 Matrix

Complete solving: \( \text{det} \begin{bmatrix} 2-\text{I} & -1 & 0 \ -1 & 2-\text{I} & -1 \ 0 & -1 & 1-\text{I} \ \right] =(2-\text{I})[(2-\text{I})(1-\text{I}) - 1] +1[-(1-\text{I})] = (2-\text{I})(2\text{I}-\text{I}^2 - 1) -1+\text{I} = (2\text{I}-\text{I}^3) - 1 + \text{I} \).
05

- Solve Determinant

Compute all terms from the previous steps in full and equate the result determinant to zero. Solve the polynomial equation: \(det(A-\text{I} \times 1)=0 \)
06

- Verify Eigenvalues

Simplify the polynomial and find all real-number solutions for eigenvalues. Ensure that \( p(\text{I}) = (\text{I}^4 - 5\text{I}^3+13\text{I}^2-18=0) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Characteristic Polynomial
To find the eigenvalues of a matrix, we first need to determine its characteristic polynomial. This polynomial is obtained from the determinant of the matrix after subtracting a scalar multiple of the identity matrix. For a matrix \( A \), this involves calculating \( \text{det}(A - \lambda I) \), where \( \lambda \) is a scalar (the eigenvalue) and \( I \) is the identity matrix of the same size as \( A \). The end goal is to find the values of \( \lambda \) that satisfy the equation \( \text{det}(A - \lambda I) = 0 \). This polynomial captures the essence of the matrix's eigenvalues.
Determinant Calculation
The determinant of a matrix is a special number that can be calculated from its elements. It is essential in finding eigenvalues and understanding matrix properties. For a 4x4 matrix, this involves breaking it down into smaller matrices whose determinants we can easily find. The determinant for a matrix \( A \), denoted as \( \text{det}(A) \), has a recursive property, meaning it can be computed via smaller 3x3, 2x2 matrices through a method called cofactor expansion. The basic principle is:
  • Choose a row or column.
  • Form minors by omitting that row and column鈥檚 elements.
  • Apply the alternating sign pattern to minors.
Cofactor Expansion
Cofactor expansion is a method to compute the determinant of a larger matrix by breaking it down into smaller matrices. Here鈥檚 a brief outline of how it works:
  • Select a row or column in the matrix to expand across. This choice can simplify calculations if a row or column has many zeros.
  • For each element in the selected row or column, form a minor by removing the corresponding row and column from the matrix.
  • Calculate the determinant of these smaller matrices (the minors).
  • Multiply each minor by the corresponding element from the original matrix and the appropriate sign from the checkerboard pattern.
  • Sum up these products to get the determinant of the original matrix.
This method helps break down complex problems into manageable pieces.
Matrix Algebra
Matrix Algebra is a system of algebra that deals with matrices. Understanding matrix algebra is crucial for solving problems involving eigenvalues and determinants. Key concepts include:
  • **Addition and Subtraction of Matrices**: This is done by adding or subtracting corresponding elements in the matrices.
  • **Matrix Multiplication**: This involves summing the products of rows and columns of two matrices.
  • **Identity Matrix**: This is a square matrix with ones on the diagonal and zeros elsewhere. It acts like the number 1 in matrix multiplication.
  • **Inverse Matrix**: A matrix that, when multiplied by its original matrix, results in the identity matrix.
These fundamental operations allow us to manipulate and solve matrix equations efficiently.

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Most popular questions from this chapter

The differential equations of motion for the mass-spring system are $$ \begin{aligned} k\left(-2 u_{1}+u_{2}\right) &=m \ddot{u}_{1} \\ k\left(u_{1}-2 u_{2}+u_{3}\right) &=3 m \ddot{u}_{2} \\ k\left(u_{2}-2 u_{3}\right) &=2 m \ddot{u}_{3} \end{aligned} $$ where \(u_{i}(t)\) is the displacement of mass \(i\) from its equilibrium position and \(k\) is the spring stiffness. Determine the circular frequencies of vibration and the corresponding mode shapes.

Compute the matrix \(\mathrm{A}^{*}\) that results from annihilation of \(A_{14}\) and \(A_{41}\) in the matrix $$ A=\left[\begin{array}{rrrr} 4 & -1 & 0 & 1 \\ -1 & 6 & -2 & 0 \\ 0 & -2 & 3 & 2 \\ 1 & 0 & 2 & 4 \end{array}\right] $$ by a Jacobi rotation.

Use Gerschgorin's theorem to determine bounds on the eigenvalues of (a) \(\mathrm{A}=\left[\begin{array}{rrr}10 & 4 & -1 \\ 4 & 2 & 3 \\ -1 & 3 & 6\end{array}\right]\) (b) \(\quad \mathbf{B}=\left[\begin{array}{rrr}4 & 2 & -2 \\ 2 & 5 & 3 \\ -2 & 3 & 4\end{array}\right]\)

Tridiagonalize the matrix $$ \mathrm{A}=\left[\begin{array}{rrr} 12 & 4 & 3 \\ 4 & 9 & 3 \\ 3 & 3 & 15 \end{array}\right] $$ with Householder's reduction.

\- Find the smallest two eigenvalues of $$ A=\left[\begin{array}{rrrr} 4 & -1 & 0 & 1 \\ -1 & 6 & -2 & 0 \\ 0 & -2 & 3 & 2 \\ 1 & 0 & 2 & 4 \end{array}\right] $$
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