Mini-Lecture 6-17 - Eigenvalues and eigenvectors.ipynb

Kernel: SageMath 9.3

import numpy as np
from ipywidgets import interactive_output, VBox, HBox

Learning goals:

• Be able to explain what an eigenvector and an eigenvalue of a matrix are.

• Be able to explain how you can see one eigenvector-eigenvalue pair in the long term behavior of a discrete-time linear model.

Recall the black bear population model:

Assumptions:

• The population is split into two types of bears: juveniles ($J$) and adults ($A$).

• Each year, on average, $42\%$ of adults give birth to a cub.

• Each year, $24\%$ of juveniles reach adulthood.

• Each year, $15\%$ of adult bears die, and $29\%$ of juvenile bears die.

Resulting model:

$$\begin{pmatrix} J_{t+1} \\ A_{t+1} \end{pmatrix} = \begin{pmatrix} 0.47 J_t + 0.42 A_t \\ 0.24 J_t + 0.85 A_t \end{pmatrix} = \begin{bmatrix} 0.47 & 0.42 \\ 0.24 & 0.85 \end{bmatrix} \begin{pmatrix} J_t \\ A_t \end{pmatrix}$$

def matrix_interactive():
    M = matrix(RDF, [[0.47, 0.42], [0.24, 0.85]])
    show(r"M = " + latex(M))

    years = 30
    solution = [vector((500.0, 250.0))]
    for t in range(years + 1):
        solution.append(M * solution[-1])

    @interact(t=slider(0, years, 1, label="$t$"))
    def update(t):
        show(LatexExpr(r"t = %d" % (t,)))
        show(LatexExpr(r"\text{Current state: } \begin{bmatrix} %.1f \\ %.1f \end{bmatrix}" % tuple(solution[t])))
        show(LatexExpr(r"\text{Next state: } \begin{bmatrix} %.1f \\ %.1f \end{bmatrix}" % tuple(solution[t + 1])))

matrix_interactive()

M = \left($$\begin{array}{rr} 0.47 & 0.42 \\ 0.24 & 0.85 \end{array}$$\right)

def blackbear_interactive_single():
    xmax, ymax = (850, 850)
    M = matrix(RDF, [[0.47, 0.42], [0.24, 0.85]])

    years = 30
    solution = [vector((500, 250))]
    for t in range(years):
        solution.append(M * solution[-1])
    solution = np.array(solution, dtype=float)

    labeltext = r"\begin{bmatrix} J_{%d} \\ A_{%d} \end{bmatrix} = \begin{bmatrix} %.1f \\ %.1f \end{bmatrix}"

    @interact(t=slider(0, years, 1, label="$t$"))
    def update(t):
        p = list_plot(solution[:t+1], color="blue")
        p.show(xmin=0, ymin=0, xmax=850, ymax=850, aspect_ratio=1, axes_labels=("$J$", "$A$"))
        show(LatexExpr(labeltext % (t, t, *solution[t])))

blackbear_interactive_single()

def blackbear_interactive_many():
    xmax, ymax = (850, 850)
    M = matrix(RDF, [[0.47, 0.42], [0.24, 0.85]])
    evector = sorted(M.eigenvectors_right())[-1][1][0]
    length = vector((xmax, ymax)).norm()

    years = 30
    t_range = srange(20)
    solutions = {
        "blue":      [vector((500, 250))], 
        "darkorange":[vector((500,  50))], 
        "red":       [vector((150, 500))], 
        "darkgreen": [vector(( 50, 700))], 
    }
    for solution in solutions.values():
        for t in range(years):
            solution.append(M * solution[-1])
    solutions = {color: np.array(solution, dtype=float) for color, solution in solutions.items()}

    controls = {color: slider(0, years, 1, label=color) for color in solutions}
    controls["show_eigenline"] = checkbox(False, label="Show “trend”")
    @interact(**controls)
    def update(**controls):
        p = Graphics()
        for color, solution in solutions.items():
            t = controls[color]
            p += list_plot(solution[:t+1], color=color)
            p += list_plot(solution[:t+1], color=color, plotjoined=True)
        if controls["show_eigenline"]:
            p += line([evector * -length, evector * length], color="magenta", thickness=3, alpha=0.6)
        p.show(xmin=0, ymin=0, xmax=850, ymax=850, aspect_ratio=1, axes_labels=("$J$", "$A$"))

blackbear_interactive_many()

def blackbear_interactive_vectors():
    xmax, ymax = (850, 850)
    M = matrix(RDF, [[0.47, 0.42], [0.24, 0.85]])

    years = 30
    t_range = srange(20)
    solution = [vector((500, 250))]
    for t in range(years + 1):
        solution.append(M * solution[-1])
    labeltext = r"\begin{bmatrix} J_{%d} \\ A_{%d} \end{bmatrix} = \begin{bmatrix} %.1f \\ %.1f \end{bmatrix}"

    @interact(t=slider(0, years, 1, label="$t$"), 
              show_vectors=checkbox(False, label="Show arrows"), 
              just2=checkbox(False, label="Just current and next"))
    def update(t, show_vectors, just2):
        p = Graphics()
        if just2:
            p += plot(solution[t], color="blue")
            p += plot(solution[t+1], color="green")
        else:
            p += list_plot(solution[:t+1], color="blue")
            if show_vectors:
                p += plot(solution[t], color="blue")
        p.show(xmin=0, ymin=0, xmax=850, ymax=850, aspect_ratio=1, axes_labels=("$J$", "$A$"))
        show(LatexExpr(labeltext % (t, t, *solution[t])))

blackbear_interactive_vectors()

So what we just said is, eventually $$[ \text{the next state} ] = ( \text{some scalar} ) [ \text{the current state} ]$$

But also, remember, we have a function $f$ for which $$[ \text{the next state} ] = f( [ \text{the current state} ] )$$

So that means that, eventually, we must have $$f( [ \text{the current state} ] ) = ( \text{some scalar} ) [ \text{the current state} ]$$

So what we just said is, eventually $$[ \text{the next state} ] = ( \text{some scalar} ) [ \text{the current state} ]$$

But also, remember, we have a matrix $M$ for which $$[ \text{the next state} ] = M [ \text{the current state} ]$$

So that means that, eventually, we must have $$M [ \text{the current state} ] = ( \text{some scalar} ) [ \text{the current state} ]$$

Definition: Given a linear function $f: \mathbb{R}^n \to \mathbb{R}^n$, an $n$-dimensional vector $\vec{v}$ is called an eigenvector for $f$ if $\vec{v} \neq 0$ and $$f(\vec{v}) = \lambda \vec{v}$$ for some scalar $\lambda$. In this case, the scalar $\lambda$ is called the eigenvalue of $f$ corresponding to eigenvector $\vec{v}$.

Definition: Given an $n \times n$ matrix $M$, an $n$-dimensional vector $\vec{v}$ is called an eigenvector for $M$ if $\vec{v} \neq 0$ and $$M \vec{v} = \lambda \vec{v}$$ for some scalar $\lambda$. In this case, the scalar $\lambda$ is called the eigenvalue of $M$ corresponding to eigenvector $\vec{v}$.

Notes:

• Here $M$ is a matrix, $\vec{v}$ is a vector, and $\lambda$ is a scalar.

• This definition means that eigenvalues and eigenvectors always go together: for any
  eigenvector of a matrix, there is a corresponding eigenvalue, and vice-versa.

• That symbol $\lambda$ is the (lowercase) Greek letter “lambda”. The prefix “eigen” ends up being applied to many things in math and science, all ultimately coming from this definition. It is of German origin, and hence is pronounced as in “Einstein”.

blackbear_interactive_vectors()

Conclusions:

1. Definition: Given an $n \times n$ matrix $M$, a vector $\vec{v}$ is called an eigenvector for $M$ with eigenvalue $\lambda$ if $\vec{v} \neq 0$ and $$M \vec{v} = \lambda \vec{v} .$$

2. Eigenvalues and eigenvectors tell us something about the long term behavior of a discrete-time linear model. But there's a lot more to this story, so we'll have to wait for more details here...