Section 13 Week 5-6 Learning Goals

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

13.1 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 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.
Compute matrix powers using the diagonalization.
Use RStudio to find complex eigenvalues and 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 \(\lambda = a + b i\).
Find the angle of rotation and the scaling factor in a \(2 \times 2\) matrix with complex eigenvalues.
Use the dominant eigenvalue and dominant eigenvector to determine the long-term behavior of a dynamical system.
Use eigenvalues to investigate a population modeled with a Leslie matrix.
Give a close-formula for a dynamical system using the eigen decomposition of a matrix

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

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 \(E_\lambda = \mathrm{Nul}(A - \lambda I)\).
A matrix is not diagonalizable when it has an eigenvalue whose algebraic mutiplicity is strictly larger than its geometrix multiplicity.
The long-term behavior of a dynamical system is determined by its dominant eigenvalue and eigenvector.
Population model predicts one of: long term growth, extinction, convergence to a stable population.