The last section demonstrated ways in which we may relate a matrix, and the effect that multiplication has on vectors, to a simpler form. For instance, if there is a basis of \(\real^n\) consisting of eigenvectors of \(A\text{,}\) we saw that \(A\) is similar to a diagonal matrix \(D\text{.}\) As a result, the effect of multiplying vectors by \(A\text{,}\) when expressed using the basis of eigenvectors, is the same as multiplying by \(D\text{.}\)
In this section, we will put these ideas to use as we explore discrete dynamical systems, first encountered in Subsection 2.5.3. Recall that we used a state vector \(\xvec\) to characterize the state of some system at a particular time, such as the distribution of delivery trucks between two locations. A matrix \(A\) described the transition of the state vector with \(A\xvec\) characterizing the state of the system at a later time. Since we would like to understand how the state vector evolves over time, we are interested in studying the sequence of vectors \(A^k\xvec\text{.}\)
Form a basis of \(\real^2\) consisting of eigenvectors of \(A\) and write the vector \(\xvec = \twovec{1}{4}\) as a linear combination of basis vectors.
We will begin with a dynamical system that illustrates how the ideas we’ve been developing can help us understand the populations of two interacting species. There are several possible ways in which two species may interact. For example, wolves on Isle Royale in northern Michigan prey on moose so this interaction is often called a predator-prey relationship. Other interactions between species, such as bees and flowering plants, are mutually beneficial for both species.
Suppose we have two species \(R\) and \(S\) that interact with each another and that we record the change in their populations from year to year. When we begin our study, the populations, measured in thousands, are \(R_0\) and \(S_0\text{;}\) after \(k\) years, the populations are \(R_k\) and \(S_k\text{.}\)
This is an example of a mutually beneficial relationship between two species. If species \(S\) is not present, then \(R_{k+1} = 0.9R_k\text{,}\) which means that the population of species \(R\) decreases every year. However, species \(R\) benefits from the presence of species \(S\text{,}\) which helps \(R\) to grow by 80% of the population of species \(S\text{.}\) In the same way, \(S\) benefits from the presence of \(R\text{.}\)
We will record the populations in a vector \(\xvec_k = \twovec{R_k}{S_k}\) and note that \(\xvec_{k+1} = A\xvec_k\) where \(A = \left[\begin{array}{rr}
0.9 \amp 0.8 \\
0.2 \amp 0.9 \\
\end{array}\right]
\text{.}\)
Suppose that initially \(\xvec_0 = \twovec{2}{3}\text{.}\) Write \(\xvec_0\) as a linear combination of the eigenvectors \(\vvec_1\) and \(\vvec_2\text{.}\)
Write the vectors \(\xvec_1\text{,}\)\(\xvec_2\text{,}\) and \(\xvec_3\) as linear combinations of the eigenvectors \(\vvec_1\) and \(\vvec_2\text{.}\)
After a very long time, by approximately what factor does the population of \(R\) grow every year? By approximately what factor does the population of \(S\) grow every year?
This activity demonstrates the type of systems we will be considering. In particular, we will have vectors \(\xvec_k\) that describe the state of the system at time \(k\) and a matrix \(A\) that describes how the state evolves from one time to the next: \(\xvec_{k+1} = A\xvec_k\text{.}\) The eigenvalues and eigenvectors of \(A\) provide the key that helps us understand how the vectors \(\xvec_k\) evolve and that enables us to make long-range predictions.
and that the matrix \(A\) has eigenvectors \(\vvec_1 =
\twovec{2}{1}\) and \(\vvec_2=\twovec{-2}{1}\) with associated eigenvalues \(\lambda_1=1.3\) and \(\lambda_2=0.5\text{.}\)
Let’s shift our perspective slightly. The eigenvectors \(\vvec_1\) and \(\vvec_2\) form a basis \(\bcal\) of \(\real^2\text{,}\) which says that \(A\) is diagonalizable; that is, \(A = PDP^{-1}\) where
The coordinate system defined by the basis \(\bcal\) can be used to express the state vectors. For instance, we can write the initial state vector \(\xvec_0=\twovec{2}{3}= 2\vvec_1+\vvec_2\text{,}\) which means that \(\coords{\xvec_0}{\bcal} = \twovec{2}{1}\text{.}\) Moreover, \(\xvec_1=A\xvec_0 = (1.3)\cdot2\vvec_1 + (0.5)\cdot1\vvec_2\) so that
Thinking about this geometrically, we begin with the vector \(\coords{\xvec_0}{\bcal}=\ctwovec{2}{1}\text{.}\) Subsequent vectors \(\coords{\xvec_k}{\bcal}\) are obtained by scaling horizontally by a factor of \(1.3\) and scaling vertically by a factor \(0.5\text{.}\) Notice how the points move along a curve away from the origin becoming ever closer to the horizontal axis. After a very long time, \(\coords{\xvec_k}{\bcal} \approx
\ctwovec{1.3^k\cdot2}{0}\text{.}\)
A plot showing how the state vector evolves in the eigenvector coordinate system. There is a standard \(1\times1\) coordinate grid and a set of axes. The first few points \(\coords{\xvec_k}{\bcal}\) are shown at roughly \((2,1)\text{,}\)\((2.6,0.5)\text{,}\)\((3.4,0.25)\text{,}\) and \((4.4,0.12)\text{.}\)
There is also a curve in the first quadrant on which all the points lie. This curve begins at the top of the diagram close to the vertical axis. As it moves down the diagram, it moves away from the vertical axis and eventually becomes close to the horizontal axis as it moves off the right side of the diagram. An arrow indicates the direction of motion.
To recover the behavior of the sequence \(\xvec_0, \xvec_1,
\xvec_2, \ldots\text{,}\) we change coordinate systems using the basis defined by \(\vvec_1\) and \(\vvec_2\text{.}\) Here, the points move along a curve away from the origin becoming ever closer to the line defined by \(\vvec_1\text{.}\)
A plot showing how the state vector evolves in the standard coordinate system. The eigenvectors \(\vvec_1\) and \(\vvec_2\) are shown along with the skewed coordinate grid representing the linear combinations of eigenvectors. The lines defined as scalar multiples of \(\vvec_1\) and \(\vvec_2\) are also indicated as a set of skewed coordinate axes.
The points \(\xvec_k\) are shown for a few values of \(k\text{.}\) For instance, \(\xvec_0 = 2\vvec_1+1\vvec_2\) appears on the skewed coordinate grid at \((2,1)\text{,}\)\(\xvec_1\) appears at \((2.6,0.5)\text{,}\) and \(\xvec_2\) at \((3.4, 0.25)\text{.}\)
There is also a curve that enters the diagram close to the skewed coordinate axis representing positive scalar multiples of \(\vvec_2\text{.}\) It moves away from that line and becomes close to the coordinate axis representing positive scalar multiples of \(\vvec_1\) as it moves off the diagram. An arrow indicates the direction of motion.
Eventually, the vectors become practically indistinguishable from a scalar multiple of \(\vvec_1 = \twovec{2}{1}\) since \(\xvec_k\approx 1.3^k\cdot2\vvec_1\text{.}\) This means that
In addition, \(\xvec_{k+1}
\approx 1.3\xvec_k\) so that \(R_{k+1}\approx 1.3 R_k\) and \(S_{k+1}\approx 1.3 S_k\text{,}\) which tells us that both populations are multiplied by 1.3 every year meaning the annual growth rate for both populations is about 30%.
In the same way, we can consider other possible initial populations \(\xvec_0\) as shown in Figure 4.4.1. Regardless of \(\xvec_0\text{,}\) the population vectors, in the coordinates defined by \(\bcal\text{,}\) are scaled horizontally by a factor of \(1.3\) and vertically by a factor of \(0.5\text{.}\) The sequence of points \(\coords{\xvec_k}{\bcal}\text{,}\) called trajectories, move along the curves, as shown on the left. In the standard coordinate system, we see that the trajectories converge to the eigenspace \(E_{1.3}\text{.}\)
On the left are shown several trajectories in the coordinate system defined by the eigenvectors. There is a standard \(1\times1\) coordinate grid and set of coordinate axes. Most trajectories are represented by a curve entering the diagram near the top or bottom of the diagram near the vertical axis and then moving away from the vertical axis and becoming close to the horizontal axis as it leaves the diagram. Two special trajectories begin on the vertical axis and move toward the origin. Two more special trajectories begin on the horizontal axis and move away from the origin. Each curve has a set of points that demonstrate the evolution of a state vector \(\coords{\xvec_k}{\bcal}\) as it is pushed along the curve.
The diagram on the right is the same only presented in the standard coordinate system. Instead of the standard coordinate grid, there is the skewed coordinate grid of lines parallel to the eigenvectors. Instead of the standard coordinate axes, there is the set of lines defined by scalar multiples of the eigenvectors. Most trajectories enter the diagram close to the scalar multiples of \(\vvec_2\) and move away from that line and toward the line defined by scalar multiples of \(\vvec_1\text{.}\) A set of points on each curve demonstrate the evolution of a state vector \(\xvec_k\) as it is pushed along the curve.
Figure4.4.1.The trajectories of the dynamical system formed by the matrix \(A\) in the coordinate system defined by \(\bcal\text{,}\) on the left, and in the standard coordinate system, on the right.
We conclude that, regardless of the initial populations, the ratio of the populations \(R_k/S_k\) will approach 2 to 1 and that the growth rate for both populations approaches 30%. This example demonstrates the power of using eigenvalues and eigenvectors to rewrite the problem in terms of a new coordinate system. By doing so, we are able to predict the long-term behavior of the populations independently of the initial populations.
Diagrams like those shown in Figure 4.4.1 are called phase portraits. On the left of Figure 4.4.1 is the phase portrait of the diagonal matrix \(D=\mattwo{1.3}00{0.5}\) while the right of that figure shows the phase portrait of \(A =
\mattwo{0.9}{0.8}{0.2}{0.9}\text{.}\) The phase portrait of \(D\) is relatively easy to understand because it is determined only by the two eigenvalues. Once we have the phase portrait of \(D\text{,}\) however, the phase portrait of \(A\) has a similar appearance with the eigenvectors \(\vvec_j\) replacing the standard basis vectors \(\evec_j\text{.}\)
In the previous example, we were able to make predictions about the behavior of trajectories \(\xvec_k=A^k\xvec_0\) by considering the eigenvalues and eigenvectors of the matrix \(A\text{.}\) The next activity looks at a collection of matrices that demonstrate the types of behavior a \(2\times2\) dynamical system can exhibit.
We will now look at several more examples of dynamical systems. If \(P = \left[\begin{array}{rr}
1 \amp -1 \\
1 \amp 1 \\
\end{array}\right]
\text{,}\) we note that the columns of \(P\) form a basis \(\bcal\) of \(\real^2\text{.}\) Given below are several matrices \(A\) written in the form \(A=PEP^{-1}\) for some matrix \(E\text{.}\) For each matrix, state the eigenvalues of \(A\) and sketch a phase portrait for the matrix \(E\) on the left and a phase portrait for \(A\) on the right. Describe the behavior of \(A^k\xvec_0\) as \(k\) becomes very large for a typical initial vector \(\xvec_0\text{.}\)
On the left is a standard \(1\times1\) coordinate grid and coordinate axes. On the right is the skewed grid and axes defined by the eigenvectors of \(A\text{.}\)
On the left is a standard \(1\times1\) coordinate grid and coordinate axes. On the right is the skewed grid and axes defined by the eigenvectors of \(A\text{.}\)
On the left is a standard \(1\times1\) coordinate grid and coordinate axes. On the right is the skewed grid and axes defined by the eigenvectors of \(A\text{.}\)
On the left is a standard \(1\times1\) coordinate grid and coordinate axes. On the right is the skewed grid and axes defined by the eigenvectors of \(A\text{.}\)
On the left is a standard \(1\times1\) coordinate grid and coordinate axes. On the right is the skewed grid and axes defined by the eigenvectors of \(A\text{.}\)
On the left is a standard \(1\times1\) coordinate grid and coordinate axes. On the right is the skewed grid and axes defined by the eigenvectors of \(A\text{.}\)
This activity demonstrates six possible types of dynamical systems, which are determined by the eigenvalues of \(A\text{.}\)
Suppose that \(A\) has two real eigenvalues \(\lambda_1\) and \(\lambda_2\) and that both \(|\lambda_1|, |\lambda_2| \gt
1\text{.}\) In this case, any nonzero vector \(\xvec_0\) forms a trajectory that moves away from the origin so we say that the origin is a repellor. This is illustrated in Figure 4.4.2.
On the left is a set of trajectories shown in the coordinate system defined by the eigenvectors. All the curves begin close to the origin and move away from the origin and out of the diagram. They move more rapidly in the vertical direction than the horizontal so they initially stay close to the vertical axis. Each curve contains a set of points that represent the evolution of a state vector as they move away from the origin.
The same diagram is presented on the right in the standard coordinate system. There is a skewed coordinate grid and axes, and a set of trajectories that begin close to the origin. The curves initially stay closer to the line defined by scalar multiples of the eigenvector corresponding to the smaller eigenvalue before moving out of the diagram. Each curve contains a set of points that move along the curve and represent the evolution of a state vector.
Suppose that \(A\) has two real eigenvalues \(\lambda_1\) and \(\lambda_2\) and that \(|\lambda_1| \gt 1 \gt |\lambda_2|
\text{.}\) In this case, most nonzero vectors \(\xvec_0\) form trajectories that converge to the eigenspace \(E_{\lambda_1}\text{.}\) In this case, we say that the origin is a saddle as illustrated in Figure 4.4.3.
On the left is a set of trajectories shown in the coordinate system defined by the eigenvectors. Each trajectory enters at the top or bottom of the diagram close to the vertical axis and moves toward the horiztonal axis exiting the diagram very close to that axis. Each curve contains a set of points that represent the evolution of a state vector as they move away from the origin.
The same diagram is presented on the right in the standard coordinate. There is a skewed coordinate grid and axes, and a set of trajectories that enter the diagram close to one of the lines defined by an eigenvector. They then move toward the line defined by the other eigenvector and exit the diagram close to that line. Each curve contains a set of points that move along the curve and represent the evolution of a state vector.
Suppose that \(A\) has two real eigenvalues \(\lambda_1\) and \(\lambda_2\) and that both \(|\lambda_1|, |\lambda_2| \lt
1\text{.}\) In this case, any nonzero vector \(\xvec_0\) forms a trajectory that moves into the origin so we say that the origin is an attractor. This is illustrated in Figure 4.4.4.
On the left is a set of trajectories shown in the coordinate system defined by the eigenvectors. Each of the curves enter the diagram from a variety of locations and then move toward the vertical axis as they eventually fall into the origin. Each curve contains a set of points that represent the evolution of a state vector as they move toward from the origin.
The same diagram is presented on the right in the standard coordinate. The curves enter the diagram from a variety of locations and get closer to the line defined by one of the eigenvectors as they fall into the origin. Each curve contains a set of points that move along the curve and represent the evolution of a state vector.
Suppose that \(A\) has a complex eigenvalue \(\lambda = a+bi\) where \(|\lambda| \gt 1\text{.}\) In this case, a nonzero vector \(\xvec_0\) forms a trajectory that spirals away from the origin. We say that the origin is a spiral repellor, as illustrated in Figure 4.4.5.
On the left is a set of trajectories shown in the coordinate system defined by the eigenvectors. Each of the curves begin near the origin but move out of the diagram as they spiral counterclockwise about the origin. Each curve contains a set of points that represent the evolution of a state vector as they move away from the origin.
The same diagram is presented on the right in the standard coordinate system. There is a skewed coordinate grid and axes, and a set of trajectories that begin close to the origin and move out of the diagram as they spiral clockwise about the origin. Each curve contains a set of points that move along the curve and represent the evolution of a state vector.
Suppose that \(A\) has a complex eigenvalue \(\lambda = a+bi\) where \(|\lambda| = 1\text{.}\) In this case, a nonzero vector \(\xvec_0\) forms a trajectory that moves on a closed curve around the origin. We say that the origin is a center, as illustrated in Figure 4.4.6.
On the left is a set of trajectories shown in the coordinate system defined by the eigenvectors. Each of the curves is a circle centered at the origin and moving counterclockwise. Each curve contains a set of points that represent the evolution of a state vector as they move away from the origin.
The same diagram is presented on the right in the standard coordinate. There is a skewed coordinate grid and axes, and a set of trajectories that move counterclockwise about the origin in ellipses. Each curve contains a set of points that move along the curve and represent the evolution of a state vector.
Suppose that \(A\) has a complex eigenvalue \(\lambda = a+bi\) where \(|\lambda| \lt 1\text{.}\) In this case, a nonzero vector \(\xvec_0\) forms a trajectory that spirals into the origin. We say that the origin is a spiral attractor, as illustrated in Figure 4.4.7.
On the left is a set of trajectories shown in the coordinate system defined by the eigenvectors. The curves enter the diagram in a variety of locations but move toward the origin as they spiral counterclockwise. Each curve contains a set of points that represent the evolution of a state vector as they move away from the origin.
The same diagram is presented on the right in the standard coordinate. The curves enter the diagram from a variety of locations but move toward the origin as they spiral counterclockwise. Each curve contains a set of points that move along the curve and represent the evolution of a state vector.
This list includes many types of expected behavior, but there are other possibilities if, for instance, one of the eigenvalues is 0. The next section explores the situation when one of the eigenvalues is 1.
In this activity, we will consider several ways in which two species might interact with one another. Throughout, we will consider two species \(R\) and \(S\) whose populations in year \(k\) form a vector \(\xvec_k=\twovec{R_k}{S_k}\) and which evolve according to the rule
Explain why the species do not interact with one another. Which of the six types of dynamical systems do we have? What happens to both species after a long time?
Explain why \(S\) is a beneficial species for \(R\text{.}\) Which of the six types of dynamical systems do we have? What happens to both species after a long time?
If \(A = \left[\begin{array}{rr}
0.7 \amp 0.5 \\
-0.4 \amp 1.6 \\
\end{array}\right]
\text{,}\) explain why this describes a predator-prey system. Which of the species is the predator and which is the prey? Which of the six types of dynamical systems do we have? What happens to both species after a long time?
Suppose that \(A = \left[\begin{array}{rr}
0.5 \amp 0.2 \\
-0.4 \amp 1.1 \\
\end{array}\right]
\text{.}\) Compare this predator-prey system to the one in the previous part. Which of the six types of dynamical systems do we have? What happens to both species after a long time?
Up to this point, we have focused on \(2\times2\) systems. In fact, the general case is quite similar. As an example, consider a \(3\times3\) system \(\xvec_{k+1}=A\xvec_k\) where the matrix \(A\) has eigenvalues \(\lambda_1 = 0.6\text{,}\)\(\lambda_2 = 0.8\text{,}\) and \(\lambda_3=1.1\text{.}\) Since the eigenvalues are real and distinct, there is a basis \(\bcal\) consisting of eigenvectors of \(A\) so we can look at the trajectories \(\coords{\xvec_k}{\bcal}\) in the coordinate system defined by \(\bcal\text{.}\) The phase portraits in Figure 4.4.8 show how some representative trajectories will evolve. We see that all the trajectories will converge into the eigenspace \(E_{1.1}\text{.}\)
Trajectories of a three dimensional dynamical system. There is the standard set of coordinate axes labelled \(x_1\text{,}\)\(x_2\text{,}\) and \(x_3\text{.}\) There is a coordinate grid representing the \(x_1x_2\)-plane. Trajectories are shown in this plane converging toward the origin as in a two dimensional attractor. A trajectory moves along the \(x_3\) axis away from the origin.
Trajectories of a three dimensional dynamical system. There is the standard set of coordinate axes labelled \(x_1\text{,}\)\(x_2\text{,}\) and \(x_3\text{.}\) There is a coordinate grid representing the \(x_1x_2\)-plane. There is a collection of trajectories that do not begin in the \(x_1x_2\)-plane or the \(x_3\) axis. These trajectories converge to the \(x_3\) axis as they move away from the origin.
Figure4.4.8.In a \(3\times3\) system with \(\lambda_1 = 0.6\text{,}\)\(\lambda_2=0.8\text{,}\) and \(\lambda_3 = 1.1\text{,}\) the trajectories \(\coords{\xvec_k}{\bcal}\) move along the curves shown above.
In the same way, suppose we have a \(3\times3\) system with complex eigenvalues \(\lambda=0.8 \pm 0.5i\) and \(\lambda_3=1.1\text{.}\) Since the complex eigenvalues satisfy \(|\lambda| \lt 1\text{,}\) there is a two-dimensional subspace in which the trajectories spiral in toward the origin. The phase portraits in Figure 4.4.9 show some of the trajectories. Once again, we see that all the trajectories converge into the eigenspace \(E_{1.1}\text{.}\)
Trajectories of a three dimensional dynamical system. There is the standard set of coordinate axes labelled \(x_1\text{,}\)\(x_2\text{,}\) and \(x_3\) and a coordinate grid representing the \(x_1x_2\)-plane. Trajectories are shown in this plane spiralling toward the origin as in a two dimensional spiral attractor. A trajectory moves along the \(x_3\) axis away from the origin.
Trajectories of a three dimensional dynamical system. There is the standard set of coordinate axes labelled \(x_1\text{,}\)\(x_2\text{,}\) and \(x_3\) and a coordinate grid representing the \(x_1x_2\)-plane. Trajectories are shown that do not begin on either the \(x_1x_2\)-plane or the \(x_3\) axis. They converge to the \(x_3\) axis by spiralling around it as they move away from the origin.
Figure4.4.9.In a \(3\times3\) system with complex eigenvalues \(\lambda = a\pm bi\) with \(|\lambda| \lt 1\) and \(\lambda_3=1.1\text{,}\) the trajectories \(\coords{\xvec_k}{\bcal}\) move along the curves shown above.
The following type of analysis has been used to study the population of a bison herd. We will divide the population of female bison into three groups: juveniles who are less than one year old; yearlings between one and two years old; and adults who are older than two years.
There are three boxes arranged in a triangle. At the top is a box labelled “Juveniles” while at the bottom left is “Adults” and at the bottom right is “Yearlings”. There is an arrow pointing from Juveniles to Yearlings that is labelled with 80%. An arrow from Yearlings to Adults is labelled 90%, and an arrow from Adults to Juveniles is labelled 40%. Finally, there is a loop from Adults back to Adults that is labelled by 80%.
As is usual, we write the matrix \(\xvec_k=\threevec{J_k}{Y_k}{A_k}\text{.}\) Write the matrix \(A\) such that \(\xvec_{k+1} = A\xvec_k\) and find its eigenvalues.
Make a prediction about the long-term behavior of \(\xvec_k\text{.}\) For instance, at what rate does it grow? For every 100 adults, how many juveniles, and yearlings are there?
Suppose that the birth rate decreases further so that only 20% of adults give birth to a juvenile. How does this affect the long-term growth rate of the herd?
We have been exploring discrete dynamical systems in which an initial state vector \(\xvec_0\) evolves over time according to the rule \(\xvec_{k+1}=A\xvec_k\text{.}\) The eigenvalues and eigenvectors of \(A\) help us understand the behavior of the state vectors. In the \(2\times2\) case, we saw that
\(|\lambda_1|, |\lambda_2| \lt 1\) produces an attractor so that trajectories are pulled in toward the origin.
\(|\lambda_1| \gt 1\) and \(|\lambda_2| \lt 1\) produces a saddle in which most trajectories are pushed away from the origin and in the direction of \(E_{\lambda_1}\text{.}\)
For each of the \(2\times2\) matrices below, find the eigenvalues and, when appropriate, the eigenvectors to classify the dynamical system \(\xvec_{k+1}=A\xvec_k\text{.}\) Use this information to sketch the phase portraits.
For the different values of \(p\text{,}\) determine which types of dynamical system results. For what range of \(p\) values do we have an attractor? For what range of \(p\) values do we have a saddle? For what value does the transition between the two types occur?
where \(p\) is a parameter. As we saw in the text, this dynamical system represents a typical predator-prey relationship, and the parameter \(p\) represents the rate at which species \(R\) preys on \(S\text{.}\) We will denote the matrix \(A=\mattwo{\frac12}{\frac12}{-p}2\text{.}\)
If \(p = 0\text{,}\) determine the eigenvectors and eigenvalues of the system and classify it as one of the six types. Sketch the phase portraits for the diagonal matrix \(D\) to which \(A\) is similar as well as the phase portrait for \(A\text{.}\)
On the left is a standard coordinate grid and axes. The right also has a standard coordinate grid and axes, but the horizontal axis is labelled \(R\) and the vertical axis \(S\text{.}\)
If \(p=1\text{,}\) determine the eigenvectors and eigenvalues of the system. Sketch the phase portraits for the diagonal matrix \(D\) to which \(A\) is similar as well as the phase portrait for \(A\text{.}\)
On the left is a standard coordinate grid and axes. The right also has a standard coordinate grid and axes, but the horizontal axis is labelled \(R\) and the vertical axis \(S\text{.}\)
Using the eigenvalues of \(A\text{,}\) we can write \(A=PEP^{-1}\) for some matrices \(E\) and \(P\text{.}\) What is the matrix \(E\) and what geometric effect does multiplication by \(E\) have on vectors in the plane?
Find a matrix \(E\) such that \(B =
PEP^{-1}\) for some matrix \(P\text{.}\) What geometric effect does multiplication by \(E\) have on vectors in the plane?
If the birth rate goes up to 80%, what prediction can you make about these populations after a very long time? For every 100 adults, how many juveniles, and yearlings are there?
Determine whether the following statements are true or false and provide a justification for your response. In each case, we are considering a dynamical system of the form \(\xvec_{k+1} =
A\xvec_k\text{.}\)
If the \(2\times2\) matrix \(A\) has a complex eigenvalue, we cannot make a prediction about the behavior of the trajectories.
If a \(4\times4\) matrix has complex eigenvalues \(\lambda_1\text{,}\)\(\lambda_2\text{,}\)\(\lambda_3\text{,}\) and \(\lambda_4\text{,}\) all of which satisfy \(|\lambda_j| \gt
1\text{,}\) then all the trajectories are pushed away from the origin.
The Fibonacci numbers form the sequence of numbers that begins \(0, 1, 1, 2, 3, 5, 8, 13, \ldots\text{.}\) If we let \(F_n\) denote the \(n^{th}\) Fibonacci number, then
The Lucas numbers \(L_n\) are defined by the same relationship as the Fibonacci numbers: \(L_{n+2}=L_{n+1}+L_n\text{.}\) However, we begin with \(L_0=2\) and \(L_1=1\text{,}\) which leads to the sequence \(2,1,3,4,7,11,\ldots\text{.}\)
As before, form the vector \(\xvec_n=\twovec{L_{n+1}}{L_n}\) so that \(\xvec_{n+1}=A\xvec_n\text{.}\) Express \(\xvec_0\) as a linear combination of \(\vvec_1\) and \(\vvec_2\text{,}\) eigenvectors of \(A\text{.}\)
Gil Strang defines the Gibonacci numbers\(G_n\) as follows. We begin with \(G_0 = 0\) and \(G_1=1\text{.}\) A subsequent Gibonacci number is the average of the two previous; that is, \(G_{n+2} = \frac12(G_{n}+G_{n+1})\text{.}\) We then have
Consider a small rodent that lives for three years. Once again, we can separate a population of females into juveniles, yearlings, and adults. Suppose that, each year,
Writing the populations of juveniles, yearlings, and adults in year \(k\) using the vector \(\xvec_k=\threevec{J_k}{Y_k}{A_k}\text{,}\) find the matrix \(A\) such that \(\xvec_{k+1} = A\xvec_k\text{.}\)
