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Chapter 4: Problem 91

Find an invertible matrix Pand a matrix \(C\) of the form \(C=\left[\begin{array}{cr}a & -b \\ b & a\end{array}\right]\) such that \(A=P C P^{-1}\) Sketch the first six points of the trajectory for the dynamical system \(\mathbf{x}_{\mathrm{k}+1}=A \mathbf{x}_{\mathrm{k}}\) with \(\mathbf{x}_{0}=\left[\begin{array}{l}1 \\ 1\end{array}\right]\) and classify the origin as \(a\) spiral attractor, spiral repeller, or orbital center $$A=\left[\begin{array}{rr} 1 & -1 \\ 1 & 0 \end{array}\right]$$

Short Answer

Expert verified
The origin is a spiral repeller with points diverging outward in a spiral trajectory.

Step by step solution

01

Verify Matrix Input

First, identify the matrix \( A \) provided in the problem statement:\[ A = \begin{bmatrix} 1 & -1 \ 1 & 0 \end{bmatrix} \] Make sure that this is the matrix to be manipulated.
02

Find Eigenvalues of A

Find the characteristic polynomial of \( A \), which is \( \lambda^2 - \lambda + 1 = 0 \). Solving this gives the eigenvalues \( \lambda = \frac{1 \pm i\sqrt{3}}{2} \). These eigenvalues indicate a spiral characteristic.
03

Find Eigenvectors of A

For each eigenvalue, solve \( (A - \lambda I)v = 0 \) to find the eigenvectors. For \( \lambda = \frac{1 + i\sqrt{3}}{2} \), the eigenvector is approximately \( v_1 = \begin{bmatrix} 1 \ \frac{3 - \sqrt{3}i}{2} \end{bmatrix} \).
04

Formulate Invertible Matrix P

Using the eigenvectors, form the matrix \( P \): \[ P = \begin{bmatrix} 1 & 1 \ \frac{3 - \sqrt{3}i}{2} & \frac{3 + \sqrt{3}i}{2} \end{bmatrix} \] The complex nature of eigenvectors implies the inverse will also contain complex numbers.
05

Find Matrix C

The matrix \( C \) is defined as \( C=\begin{bmatrix}a & -b \ b & a \end{bmatrix} \). Set \( a = \text{Re}(\lambda) = \frac{1}{2} \) and \( b = \text{Im}(\lambda) = \frac{\sqrt{3}}{2} \), giving \[ C = \begin{bmatrix} \frac{1}{2} & -\frac{\sqrt{3}}{2} \ \frac{\sqrt{3}}{2} & \frac{1}{2} \end{bmatrix} \].
06

Verify Decomposition A = PCP^{-1}

Substitute \( P \), \( C \), and their inverse into the equation \( A = P C P^{-1} \) to verify the decomposition yields the original matrix \( A \). Ensure all matrix multiplications lead back to \( A \).
07

Solving the Trajectory

Compute \( \mathbf{x}_{k+1} = A \mathbf{x}_k \) iteratively beginning with \( \mathbf{x}_0 = \begin{bmatrix} 1 \ 1 \end{bmatrix} \). Calculate \( \mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_6 \) points by continuing this iteration.
08

Classify the Origin

Because the eigenvalues have real parts equal to \( \frac{1}{2} \), the origin is a spiral repeller since they are outside the unit circle (indicating divergence).

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Key Concepts

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

Invertible Matrix
An invertible matrix, also known as a non-singular or non-degenerate matrix, is a square matrix that has an inverse. This means that there exists another matrix, say \( A^{-1} \), such that when it is multiplied by the original matrix \( A \), it yields the identity matrix. Mathematically, this is expressed as:\[ A \cdot A^{-1} = I \]where \( I \) is the identity matrix.
  • For a matrix to be invertible, its determinant must be non-zero.
  • An invertible matrix allows us to solve systems of linear equations by matrix algebra.
  • Finding an invertible matrix \( P \) is essential in matrix decomposition tasks, such as solving the decomposition \( A = P C P^{-1} \).
In the context of the exercise, the matrix \( P \) was formed using eigenvectors of \( A \), ensuring it is invertible for further calculations.
Dynamical System
A dynamical system is a mathematical framework used to describe the evolution of a point in space over time. It involves state variables that capture the state of the system at any given time. A time evolution rule, which is given by mathematical functions or equations, specifies how these state variables change over time.
  • Linear dynamical systems often use matrices to represent state transformations.
  • The solution to a linear dynamical system can demonstrate behaviors such as attraction, repulsion, or oscillation depending on the eigenvalues of the associated matrix \( A \).
  • In this exercise, the equation \( \mathbf{x}_{k+1} = A \mathbf{x}_k \) defines the dynamical system, indicating how the state evolves.
The classification of whether the origin is a spiral attractor, repeller, or orbital center is critical in the analysis of these systems, guiding us through their long-term behavior.
Characteristic Polynomial
The characteristic polynomial is a crucial concept in linear algebra, used to determine the eigenvalues of a matrix. Given a square matrix \( A \), the characteristic polynomial is derived from the determinant of \( (A - \lambda I) \), where \( \lambda \) represents eigenvalues, and \( I \) is the identity matrix.
  • Solving the characteristic polynomial gives the eigenvalues of the matrix.
  • The roots of the characteristic polynomial indicate potential directions or transformations in space.
  • For the matrix \( A \) in this exercise, the characteristic polynomial was \( \lambda^2 - \lambda + 1 = 0 \).
By solving this polynomial, the resulting eigenvalues help us understand the nature of the dynamical system, such as identifying spiral characteristics due to complex eigenvalues.
Matrix Decomposition
Matrix decomposition involves breaking down a matrix into several constituent components that are easier to work with, particularly for solving systems of equations, computing matrix powers, or inverting matrices.
  • The primary goal is to express a matrix in terms of simpler, well-defined matrices.
  • In this exercise, \( A = PCP^{-1} \) is a type of decomposition, where the invertible matrix \( P \) and a simpler matrix \( C \) are found.
  • Matrix decompositions reveal properties like stability and facilitate computations in control systems and computer graphics.
The matrix \( C \) used here has a special form showing orthogonal transformations, further illustrating universal decomposition techniques applicable across diverse mathematical and engineering applications.

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Most popular questions from this chapter

It can be shown that a nonnegative \(n \times n\) matrix is irreducible if and only if \((I+A)^{n-1}>0 .\) Use this criterion to determine whether the matrix \(A\) is irreducible. If \(A\) is reducible, find a permutation of its rows and columns that puts \(A\) into the block form \\[ \left[\begin{array}{ll} B & C \\ O & D \end{array}\right] \\] $$A=\left[\begin{array}{lllll} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 \end{array}\right]$$

If \(A\) is an invertible \(n \times n\) matrix, show that adj \(A\) is also invertible and that \\[ (\operatorname{adj} A)^{-1}=\frac{1}{\operatorname{det} A} A=\operatorname{adj}\left(A^{-1}\right) \\]

Consider the dynamical system \(\mathbf{x}_{k+1}=A \mathbf{x}_{k}\) (a) Compute and plot \(\mathbf{x}_{0}, \mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{x}_{3}\) for \(\mathbf{x}_{0}=\left[\begin{array}{l}1 \\\ 1\end{array}\right]\) (b) Compute and plot \(\mathbf{x}_{0}, \mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{x}_{3}\) for \(\mathbf{x}_{0}=\left[\begin{array}{l}1 \\\ 0\end{array}\right]\) (c) Using eigenvalues and eigenvectors, classify the origin as an attractor, repeller, saddle point, or none of these. (d) Sketch several typical trajectories of the system. $$A=\left[\begin{array}{ll} 2 & 1 \\ 0 & 3 \end{array}\right]$$

Find the general solution to the given system of differential equations. Then find the specific solution that satisfies the initial conditions. (Consider all functions to be functions of \(t .\) $$\begin{array}{ll} x_{1}^{\prime}=x_{1}+x_{2}, & x_{1}(0)=1 \\ x_{2}^{\prime}=x_{1}-x_{2}, & x_{2}(0)=0 \end{array}$$

Suppose that \(A\) is a \(6 \times 6\) matrix with characteristic polynomial \(c_{A}(\lambda)=(1+\lambda)(1-\lambda)^{2}(2-\lambda)^{3}\) (a) Prove that it is not possible to find three linearly independent vectors \(\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\) in \(\mathbb{R}^{6}\) such that \(A \mathbf{v}_{1}=\mathbf{v}_{1}, A \mathbf{v}_{2}=\mathbf{v}_{2},\) and \(A \mathbf{v}_{3}=\mathbf{v}_{3}\) (b) If \(A\) is diagonalizable, what are the dimensions of the eigenspaces \(E_{-1}, E_{1},\) and \(E_{2} ?\)
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