Vector 14 Week 5-6 Learning Goals

Here are the knowledge and skills you should master by the end of the fifth and sixth weeks.

14.1 Eigensystems

I should be able to do the following tasks:

  • Check whether a given vector \(\mathsf{v}\) is an eigenvector for square matrix \(A\).
  • Find the eigenvalues of a matrix \(2 \times 2\) matrix by hand, using the characteristic equation
  • Find the eigenvalues of a triangular matix by inspection.
  • Given the eigenvalues of matrix \(A\), find the eigenvectors by solving \((A \lambda I) = \mathbf{0}\).
  • Find the eigenvalues and (“human readable”) eigenvectors of an \(n \times n\) matrix \(A\) using eigen(A) on RStudio.
  • Determine whether a matrix is diagonalizable.
  • Factor a diagonalizable \(n \times n\) matrix as \(A = PDP^{-1}\) where \(D\) is a diagonal matrix of eigenvalues and \(P\) is the matrix whose columns are the corresponding eignvectors.
  • Use RStudio to find complex eigenvalues and (“human readable”) eigenvectors of a square matrix.
  • Factor a \(2 \times 2\) scaling-rotation matrix as \(A = P C P^{-1}\) where \(C\) is a scaling-rotation matrix \(\begin{bmatrix} a & -b \\ b & a \end{bmatrix}\) and \(P = [ \mathsf{w}, \mathsf{u}]\) where \(\mathsf{v} = \mathsf{u} + i \mathsf{w}\) is the eigenvector for \(\lambda = a + b i\).
  • Use RStudio to find the Gould Index of a network
  • Use RStudio to create a 2D plot of pairs of eigenvalues of a square matrix
  • Use the dominant eigenvalue and dominant eigenvector to determine the long-term behavior of a dynamical system
  • Create a trajectory of a \(2 \times 2\) dynamical system (either using RStudio or by using a given vector field plot) and then relate the trajectory to the eigenvectors and eigenvalues
  • Interpret the constants in the \(2 \times 2\) matrix for two interacting populations (competition, predator-prey, mutualism, etc)
  • Use RStudio to investigate the animal population modeled with a Leslie matrix.

14.2 Vocabulary

I should know and be able to use and explain the following terms or properties.

  • eigenvalue, eigenvector and eigenspace
  • characteristic equation
  • diagonalizable matrix
  • similar matrices
  • algebraic multiplicity of an eigenvalue
  • geometric multiplicity of an eigenvalue
  • scaling-rotation matrix
  • Gould Index
  • discrete dynamical system
  • trajectory
  • dominant eigenvalue and dominant eigenvector
  • population model
  • Leslie matrix

14.3 Conceptual Thinking

I should understand and be able to explain the following concepts:

  • An eigenspace of \(A\) is a subspace that is fixed under the linear transformation \(T(\mathsf{x}) = A \mathsf{x}\).
  • An eigenvalue \(\lambda\) with \(1 <| \lambda |\) corresponds to expansion.
  • An eigenvalue \(\lambda\) with \(0 < | \lambda | < 1\) corresponds to contraction.
  • A complex eigenvalue corresponds to a rotation in a 2D subspace.
  • The eigenspace for \(\lambda\) is the subspace \(\mathrm{Nul}(A - \lambda I)\).
  • A matrix is not diagonalizable when it has complex eigenvalues.
  • A matrix is not diagonalizable when it has an eigenvalue whose algebraic mutiplicity is strictly larger than its geometrix multiplicity.
  • The Gould Index measures the centrality of a vertex in the network.
  • The eigenvalues of a matrix “encode some of the patterns” found in the matrix. Larger magnitude eigenvalues indicate more important patterns.
  • The long-term behavior of a dynamical system is determined by its dominent eigenvalue and eigenvector.
  • Any population model predicts one of: long term growth, extinction, convergence to a stable population, or convergence to a stable orbit of populations.