Eigenvalues & Eigenvectors + applications: notes and practice questions

A local area has two popular coffee shops, 'The Daily Grind' (D) and 'Morning Brew' (M). A market research study models customer loyalty using a transition matrix. It is found that if a customer visited 'The Daily Grind' today, there is a 70% chance they will visit 'The Daily Grind' tomorrow and a 30% chance they will switch to 'Morning Brew'. If a customer visited 'Morning Brew' today, there is a 40% chance they will switch to 'The Daily Grind' tomorrow and a 60% chance they will stay with 'Morning Brew'.

The transition matrix $T$ for this system is given by $T = \begin{pmatrix} 0.7 & 0.4 \\ 0.3 & 0.6 \end{pmatrix}$. The matrix $T$ has eigenvalues $1$ and $0.3$.

Find an eigenvector corresponding to the eigenvalue of $1$. Give your answer in the form $\begin{pmatrix} a \\ b \end{pmatrix}$, where $a, b \in \mathbb{Z}$.

Using your answer to (a), or otherwise, find the long-term probability of a customer visiting 'The Daily Grind'. Give your answer in the form $\frac{c}{d}$, where $c, d \in \mathbb{Z}^+$.

A new streaming service, CineStream (C), enters the market, competing with the established service, FilmFlick (F).

Market research shows that each month:

• 10% of CineStream subscribers switch to FilmFlick.

• 30% of FilmFlick subscribers switch to CineStream.

The transition matrix $T$ describing these changes, where the rows and columns are ordered (CineStream, FilmFlick), is given as $T = \begin{pmatrix} 0.90 & 0.30 \\ 0.10 & 0.70 \end{pmatrix}$.

The two eigenvalues for this matrix are $1$ and $0.60$. An eigenvector corresponding to the eigenvalue of $1$ is $\begin{pmatrix} 3 \\ 1 \end{pmatrix}$.

Find an eigenvector corresponding to the eigenvalue of $0.60$.

A diagonal matrix of eigenvalues is $D = \begin{pmatrix} 0.60 & 0 \\ 0 & 1 \end{pmatrix}$.

Write down an expression for $D^n$, giving your answer as a $2 \times 2$ matrix in terms of $n$.

When CineStream and FilmFlick first launched, there were a total of $8000$ users, all of whom initially subscribed to CineStream.

Assuming the total number of users remains constant, find an expression for the number of users who will favour FilmFlick after $n$ months.

A linear transformation $T$ maps points in the plane such that a point $(x, y)$ is transformed to $(x', y')$ by the matrix $M = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix}$.

(a) Find the eigenvalues and corresponding eigenvectors of $M$.

(b) Hence find a matrix $P$ and a matrix $D$ such that $D = P^{-1}MP$.

(c) Find a general expression for $M^n$ in terms of $n$.

A scientific experiment involves a platform that oscillates with damped motion. The displacement, $x$, of the platform, measured in centimetres from its equilibrium position, can be modelled by the second order differential equation:

$\ddot{x} + 5\dot{x} + 4x = 0$, where $t$ is the time in seconds after the initial displacement.

Given that $y = \dot{x}$, show that $\dot{y} = -4x - 5y$.

The differential equation can be expressed in the form $\begin{pmatrix} \dot{x} \\ \dot{y} \end{pmatrix} = A \begin{pmatrix} x \\ y \end{pmatrix}$, where A is a $2 \times 2$ matrix.

Write down the matrix A.

Find the eigenvalues of matrix A.

Find the eigenvectors of matrix A.

Given that at $t = 0$, the platform is displaced $6$ cm from equilibrium and its velocity is $3$ cm/s, find an expression for $x$ in terms of $t$.

The populations of two interacting species, A and B, in a controlled environment are modelled by the following system of linear differential equations:

$\dot{x} = x + 2y$

$\dot{y} = 3x + 2y$

where $x(t)$ represents the population of species A and $y(t)$ represents the population of species B at time $t$. Find the general solution for this system.

Sketch the phase portrait for the system, clearly indicating the critical point, the equations of any asymptotes, and the general direction of trajectories. Briefly describe the long-term behaviour of the populations.

The concentration of a chemical, $C$ (in mol dm$^{-3}$), in a reaction vessel at time $t$ seconds is modelled by the differential equation

$\frac{\mathrm{d}^2C}{\mathrm{d}t^2} + 7\frac{\mathrm{d}C}{\mathrm{d}t} + 10C = 0$

(a) (i) Use the substitution $V = \frac{\mathrm{d}C}{\mathrm{d}t}$ to show that this equation can be written as

$\begin{pmatrix} \frac{\mathrm{d}C}{\mathrm{d}t} \\ \frac{\mathrm{d}V}{\mathrm{d}t} \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -10 & -7 \end{pmatrix} \begin{pmatrix} C \\ V \end{pmatrix}$.

(ii) Find the eigenvalues for the matrix $\begin{pmatrix} 0 & 1 \\ -10 & -7 \end{pmatrix}$.

(iii) Hence state the long-term rate of change of the chemical concentration.

The chemical reaction is now subjected to an external input, and the equation for the concentration is amended to

$\frac{\mathrm{d}^2C}{\mathrm{d}t^2} + 7\frac{\mathrm{d}C}{\mathrm{d}t} + 10C = 2t + 5$.

(b) (i) Use the substitution $V = \frac{\mathrm{d}C}{\mathrm{d}t}$ to write the differential equation as a system of coupled, first order differential equations. State the initial conditions given that, at $t = 0$, the concentration of chemical C is $1$ mol dm$^{-3}$ and its rate of change is $0$ mol dm$^{-3}$ s$^{-1}$.

(ii) Use Euler's method with a step length of $0.1$ to find the concentration of the chemical when $t = 1$ s. Give your answer to three significant figures.

(iii) Find the long-term rate of change of the chemical concentration.

In a chemical reactor, the concentrations of two reactants, Chemical A ($C_A$) and Chemical B ($C_B$), in mol/L, change over time $t$ in hours. The rate of change can be modelled by the coupled differential equations:

$C_A'(t) = C_B$

$C_B'(t) = 2C_A - C_B$

Initially, the concentration of Chemical A is $10$ mol/L and Chemical B is $5$ mol/L.

Use the matrix-eigenvalue method to find the solution for $C_A(t)$ and $C_B(t)$.

Calculate the concentrations of Chemical A and Chemical B after $0.5$ hours, giving your answers to $3$ significant figures.

As $t \to \infty$, give a simpler approximation for the concentration of Chemical A.

State what happens to the concentration of Chemical B as $t \to \infty$.

A chemical engineer is studying the concentration of an intermediate product, $C$, in a reaction vessel. The change in concentration over time $t$ (in minutes) is modelled by the second order differential equation:

$\frac{d^2C}{dt^2} + 4\frac{dC}{dt} + 3C = 0$

where $t \ge 0$. It is known that when $t = 0$, the initial concentration is $C = 0$ and the rate of change of concentration is $\frac{dC}{dt} = 2$.

Show that the system of coupled first order equations:

$\frac{dC}{dt} = y$

$\frac{dy}{dt} = -3C - 4y$

can be written as the given second order differential equation.

Find the eigenvalues of the system of coupled first order equations given in part (a).

Hence find the exact solution of the second order differential equation, given the initial conditions $C(0) = 0$ and $\frac{dC}{dt}(0) = 2$.

Sketch the graph of $C$ against $t$ for $t \ge 0$, labelling the maximum point of the graph with its coordinates.

If the concentration of the intermediate product $C$ exceeds $0.2$ arbitrary units, the reaction needs to be monitored closely. Use the model to calculate the total amount of time (in minutes) during which the reaction needs to be monitored closely.

The chemical engineer decides to monitor the reaction for 15% longer than the time found from the model in part (e).

Write down one reason, with reference to the context, to support this decision.

(a) A specialized sensor's response to a sudden input is modeled by the second-order differential equation

$\frac{d^2x}{dt^2} + 6\frac{dx}{dt} + 8x = 0$

Formulate this second-order differential equation as a system of two coupled first-order differential equations in the form $\frac{d\mathbf{X}}{dt} = A\mathbf{X}$, where $\mathbf{X} = \begin{pmatrix} x \\ y \end{pmatrix}$ and $y = \frac{dx}{dt}$.

(b) Find the eigenvalues of the coefficient matrix $A$ from part (a).

(c) Find the eigenvectors corresponding to each eigenvalue found in part (b).

(d) Write down the general solution for $x(t)$ for the given differential equation.

(a) A student's study habits are modelled by a Markov chain. If the student studies Mathematics (M) on a given day, the probability they study Mathematics the following day is $0.75$. If the student studies Physics (P) on a given day, the probability they study Physics the following day is $0.85$.

Write down a transition matrix, $T$, that shows the movement of the student's study focus between Mathematics and Physics.

(b) On Monday, a student spent their study time on Mathematics. Find the probability that the student will be studying Physics on Friday.

(c) Write down the characteristic polynomial for the matrix $T$. Give your answer in the form $\lambda^2 + b\lambda + c = 0$.

(d) Calculate the eigenvectors for the matrix $T$.

(e) Write down matrices $P$ and $D$ such that $T= PDP^{-1}$, where $D$ is a diagonal matrix.

(f) Hence, find the long-term probability that the student is studying Physics.

A financial analyst models the annual change in the value of two investment portfolios, Portfolio A and Portfolio B, using a transition matrix $M$. The matrix $M$ is given by $M = \begin{pmatrix} 1 & 2 \\ 3 & 0 \end{pmatrix}$.

(a) Find the eigenvalues of $M$.

(b) The matrix $M$ can be diagonalized such that $M = PDP^{-1}$, where $D$ is a diagonal matrix.

(i) Write down the matrix $D$.

(ii) Find the matrix $P$.

On any given day, the probability that a customer buys a particular product depends only on which product they bought the previous day.

If a customer bought Product A on the previous day, the probability they buy Product A again today is $0.7$.

If a customer bought Product B on the previous day, the probability they buy Product A today is $q$.

On day $n$ this can be represented using the vector $v_n$ where

$v_n = \begin{pmatrix} \text{probability that a customer buys Product A on day } n \\ \text{probability that a customer buys Product B on day } n \end{pmatrix}$

A Markov chain model is formed where

$v_{n+1} = Mv_n$

Matrix $M$ is of the form $\begin{pmatrix} a & q \\ b & 1-q \end{pmatrix}$

Write down the value of

(i) $a$.

Write down the value of

(ii) $b$.

On day zero, a customer buys Product A. Find the probability

(i) that the customer buys Product A for all days from $n = 1$ to $n = 4$.

On day zero, a customer buys Product A. Find the probability

(ii) that the customer buys Product A on day 4, when $q = 0.2$.

Demonstrate that, for all values of $q$, one eigenvector of $M$ is $\begin{pmatrix} 1 \\ -1 \end{pmatrix}$ and hence state the associated eigenvalue.

Find, in terms of $q$, the steady state probability that a customer buys Product A on a given day.

In the long term, the company wants Product A to have at least 70% market share.

Find the minimum value of $q$ required for this to occur.

A financial analyst models the daily price movement of a tech company's stock using a transition matrix. If the stock price increases on a particular day, the probability that it will increase the following day is $0.7$. If the stock price decreases on a particular day, the probability that it will increase the following day is $0.4$.

The transition matrix $T$ for this model is given by $T = \begin{pmatrix} 0.7 & 0.4 \\ 0.3 & 0.6 \end{pmatrix}$, where the rows/columns represent 'Increase' and 'Decrease' respectively.

(a) Given that the stock price increased today, calculate the probability that the stock price will increase in two days' time.

(b) Find the eigenvalues and corresponding eigenvectors of $T$.

(c) The matrix $T$ can be written in the form $PDP^{-1}$, where $D$ is a diagonal matrix.

(i) Write down a possible matrix $P$.

(ii) Write down the corresponding matrix $D$.

(d) Hence, determine the long-term percentage of days that the stock price will increase.

A competitive market has two dominant smartphone brands, Aura and Zenith. Each year, it is observed that $15\%$ of Aura customers switch to Zenith, while $5\%$ of Zenith customers switch to Aura. All other customer movements are negligible.

Write down a transition matrix $T$ representing the customer movements between the two brands in a particular year. Assume the order of brands is Aura, then Zenith.

Find the eigenvalues and corresponding eigenvectors of $T$.

Hence write down matrices $P$ and $D$ such that $T = PDP^{-1}$.

Initially, Brand Aura has $100\,000$ customers and Brand Zenith has $200\,000$ customers.

Find an expression for the number of customers Brand Aura has after $n$ years, where $n \in \mathbb{N}$.

Hence write down the number of customers that Brand Aura can expect to have in the long term.

A market research firm models customer loyalty between two competing coffee shops, "The Daily Grind" (D) and "Bean There, Done That" (B), using a Markov chain. Each day, a customer may switch between the two shops.

The model is of the form

$\begin{pmatrix} D_{n+1} \\ B_{n+1} \end{pmatrix} = M \begin{pmatrix} D_n \\ B_n \end{pmatrix}$

where $D_n$ is the probability a customer chooses "The Daily Grind" on day $n$, and $B_n$ is the probability a customer chooses "Bean There, Done That" on day $n$, where $n \in \mathbb{N}$.

The transition matrix $M$ is found to be $\begin{pmatrix} 0.85 & k \\ 0.15 & 0.90 \end{pmatrix}$.

Write down the value of $k$.

State what $k$ represents in this context.

Find the eigenvalues of $M$.

Find the eigenvectors of $M$.

A new customer initially chooses 'The Daily Grind'. Calculate the probability that this customer chooses 'The Daily Grind' after 3 days.

Calculate the probability that this customer chooses 'The Daily Grind' in the long term.

The following question explores a possible method of drawing phase portraits for non-linear coupled systems, taking a predator-prey model as a particular example.

A forest ecosystem contains a population of rabbits ($x$, measured in hundreds), and a population of wolves ($y$, measured in hundreds).

Research indicates that the population dynamics of both rabbits and wolves can be modelled by the following differential equations, in which $t$ is measured in years.

$\frac{dx}{dt} = 3x - \frac{xy}{2}$

$\frac{dy}{dt} = -2y + \frac{2xy}{5}$

for $x, y \ge 0$

At a specific time, there are 400 rabbits and 400 wolves, represented here by the coordinate pair (4, 4). At this time, determine the rate of change of

rabbits.

wolves.

There are two equilibrium points for the populations: A(0, 0) and B ($p, q$).

Explain why A is an equilibrium point.

Find the value of $p$ and the value of $q$.

At points close to A(0, 0), we can ignore the $xy$ terms, so that the system can be approximated by:

$\frac{dx}{dt} = 3x$

$\frac{dy}{dt} = -2y$

for $x, y \ge 0$.

By solving these two differential equations,

find an expression for $x$ in terms of $t$.

find an expression for $y$ in terms of $t$.

Using your answers from part (c), show that phase portrait trajectories close to A may be given by the equation $x^2y^3 = k$, where $k$ is a positive constant.

Hence sketch, on a phase portrait, one possible trajectory for small values of $x$ and $y$.

Now consider points $(x, y)$ close to B on the phase plane. These coordinates can be rewritten as $x = p + X$ and $y = q + Y$, where $p$ and $q$ are the values from part (b)(ii).

By substituting into the original model, show that, for small values of $X$ and $Y$:

$\dot{X} \approx -\frac{5Y}{2}$

Similarly, it can be shown that $\dot{Y} \approx \frac{12X}{5}$.

Given that $\begin{pmatrix} \dot{X} \\ \dot{Y} \end{pmatrix} = M \begin{pmatrix} X \\ Y \end{pmatrix}$, where $M$ is a square matrix, write down $M$.

By finding the eigenvalues of $M$, describe the path of the trajectories close to point B.

Hence sketch a complete set of trajectories in the phase plane for the original model, clearly indicating both equilibrium points.

In this forest ecosystem, at a specific time, there are 400 rabbits and 400 wolves.

Based on the values found in part (a), the wildlife keeper is worried and assumes that the wolves will quickly die out. Suggest whether this assumption is supported by the model. Justify your answer.

If a mechanical arm attached to a rotating system moves along a path is influenced by frictional forces. Let $x$ represent the position (in meters) of a point on the arm, measured from a fixed origin. If the motion of $x$ is modelled through the following second order differential equation:

$\ddot{x} + 5\dot{x} + 4x = 0$

Where $t$ is the time in seconds.

$a$ Take $\dot{x} = y$ and find an expression for $\dot{y}$ in terms of $x$ and $y$.

$b$ Find $A$ such that $\begin{pmatrix} \dot{x} \\ \dot{y} \end{pmatrix} = A\begin{pmatrix} x \\ y \end{pmatrix}$, where $A$ is a $2 \times 2$ matrix.

$c$ Find the eigenvalues and eigen vectors of matrix $A$.

$d$ If at $t = 0,x = 0$ and $\dot{x} = 2$ find the exact solution of this differential equation.

If a particle moves in a magnetic field and its motion is described by its displacement $x$ (in meters) from equilibrium position. If the magnetic force affects the motion $x$ of the particle and leads to the following second order differential equation:

$\ddot{x} + 6\dot{x} + 8x = 0$

Where $t$ is time in seconds.

$a$ Take $\dot{x} = y$ and find an expression for $\dot{y}$ in terms of $x$ and $y$.

$b$ Find $A$ such that $\begin{pmatrix} \dot{x} \\ \dot{y} \end{pmatrix} = A\begin{pmatrix} x \\ y \end{pmatrix}$, where $A$ is a $2 \times 2$ matrix.

$c$ Find the eigenvalues and eigen vectors of matrix $A$.

$d$ Determine whether the equilibrium point (0,0) is stable or unstable.

$e$ Find $\frac{dy}{dx}$ at the point (1,2) and (3,2).

If the equation is amended to the following:

$\ddot{x} + 6\dot{x} + 8x = 5t + 1$

$f$ At $t = 0$ take $x = 0$ cm and $\dot{x} = 0$. Use Euler's method with a step length of 0.25 to find the value of $x$ at $t = 1$.

A marketing analyst models the monthly customer loyalty for two companies, Innovate Inc. and Legacy Corp., using the transition matrix $T = \begin{pmatrix} p & 1-q \\ 1-p & q \end{pmatrix}$, where $0 < p < 1$ and $0 < q < 1$. `p` is the probability a customer of Innovate Inc. remains with them the next month, and `q` is the probability a customer of Legacy Corp. remains with them the next month.

(a) Find the eigenvalues of the matrix T in terms of `p` and `q`.

(b) In the long term, the market share for the two companies reaches a steady state. Find the proportion of customers that are with Innovate Inc. and Legacy Corp. at this steady state, in terms of `p` and `q`.

Two coffee shops, Bean Buzz and Daily Grind, compete for customers in a small town. A market analysis shows that customer loyalty changes from month to month.

A customer who went to Bean Buzz in one month has a 60% probability of switching to Daily Grind the next month. A customer who went to Daily Grind in one month has a 30% probability of switching to Bean Buzz the next month.

This situation can be modelled by the transition matrix $T=\begin{pmatrix} 0.4 & 0.3 \\ 0.6 & 0.7 \end{pmatrix}$, where the first column represents the initial state of being a Bean Buzz customer and the second column represents the initial state of being a Daily Grind customer. The eigenvalues of $T$ are $1$ and $0.1$.

(a) Find an eigenvector corresponding to the eigenvalue of $1$. Give your answer in the form $\begin{pmatrix} x \\ y \end{pmatrix}$, where $x, y \in \mathbb{Z}^+$.

(b) Using your answer to part (a), or otherwise, find the long-term market share for Bean Buzz. Give your answer as a percentage to one decimal place.