5.2: Parasitic Helminths
- Explain why we include the study of parasitic worms within the discipline of microbiology
- Compare the basic morphology of the major groups of parasitic helminthes
- Describe the characteristics of parasitic nematodes, and give an example of infective eggs and infective larvae
- Describe the characteristics of parasitic trematodes and cestodes, and give examples of each
- Identify examples of the primary causes of infections due to nematodes, trematodes, and cestodes
- Classify parasitic worms according to major groups
Parasitic helminths are animals that are often included within the study of microbiology because many species of these worms are identified by their microscopic eggs and larvae. There are two major groups of parasitic helminths: the roundworms (Nematoda) and flatworms (Platyhelminthes). Of the many species that exist in these groups, about half are parasitic and some are important human pathogens. As animals, they are multicellular and have organ systems. However, the parasitic species often have limited digestive tracts, nervous systems, and locomotor abilities. Parasitic forms may have complex reproductive cycles with several different life stages and more than one type of host. Some are monoecious, having both male and female reproductive organs in a single individual, while others are dioecious, each having either male or female reproductive organs.
Nematoda (Roundworms)
Phylum Nematoda (the roundworms) is a diverse group containing more than 15,000 species, of which several are important human parasites (Figure \(\PageIndex{1}\)). These unsegmented worms have a full digestive system even when parasitic. Some are common intestinal parasites, and their eggs can sometimes be identified in feces or around the anus of infected individuals. Ascaris lumbricoides is the largest nematode intestinal parasite found in humans; females may reach lengths greater than 1 meter. A. lumbricoides is also very widespread, even in developed nations, although it is now a relatively uncommon problem in the United States. It may cause symptoms ranging from relatively mild (such as a cough and mild abdominal pain) to severe (such as intestinal blockage and impaired growth).
Of all nematode infections in the United States, pinworm (caused by Enterobius vermicularis ) is the most common. Pinworm causes sleeplessness and itching around the anus, where the female worms lay their eggs during the night. Toxocara canis and T. cati are nematodes found in dogs and cats, respectively, that can be transmitted to humans, causing toxocariasis. Antibodies to these parasites have been found in approximately 13.9% of the U.S. population, suggesting that exposure is common. 1 Infection can cause larval migrans, which can result in vision loss and eye inflammation, or fever, fatigue, coughing, and abdominal pain, depending on whether the organism infects the eye or the viscera. Another common nematode infection is hookworm, which is caused by Necator americanus (the New World or North American hookworm) and Ancylostoma duodenale (the Old World hookworm). Symptoms of hookworm infection can include abdominal pain, diarrhea, loss of appetite, weight loss, fatigue, and anemia.
Trichinellosis, also called trichinosis, caused by Trichinella spiralis , is contracted by consuming undercooked meat, which releases the larvae and allows them to encyst in muscles. Infection can cause fever, muscle pains, and digestive system problems; severe infections can lead to lack of coordination, breathing and heart problems, and even death. Finally, heartworm in dogs and other animals is caused by the nematode Dirofilaria immitis , which is transmitted by mosquitoes. Symptoms include fatigue and cough; when left untreated, death may result.
The physician explains to Sarah’s mother that ringworm can be transferred between people through touch. “It’s common in school children, because they often come in close contact with each other, but anyone can become infected,” he adds. “Because you can transfer it through objects, locker rooms and public pools are also a potential source of infection. It’s very common among wrestlers and athletes in other contact sports.”
Looking very uncomfortable, Sarah says to her mother “I want this worm out of me.”
The doctor laughs and says, “Sarah, you’re in luck because ringworm is just a name; it is not an actual worm. You have nothing wriggling around under your skin.”
“Then what is it?” asks Sarah.
- What type of pathogen causes ringworm?
- What is the most common nematode infection in the United States?
Platyhelminths (Flatworms)
Phylum Platyhelminthes (the platyhelminths) are flatworms. This group includes the flukes, tapeworms, and the turbellarians, which include planarians. The flukes and tapeworms are medically important parasites (Figure \(\PageIndex{2}\)).
The flukes (trematodes) are nonsegmented flatworms that have an oral sucker (Figure \(\PageIndex{3}\)) (and sometimes a second ventral sucker) and attach to the inner walls of intestines, lungs, large blood vessels, or the liver. Trematodes have complex life cycles, often with multiple hosts. Several important examples are the liver flukes ( Clonorchis and Opisthorchis ), the intestinal fluke ( Fasciolopsis buski ), and the oriental lung fluke ( Paragonimus westermani ). Schistosomiasis is a serious parasitic disease, considered second in the scale of its impact on human populations only to malaria. The parasites Schistosoma mansoni, S. haematobium , and S. japonicum , which are found in freshwater snails, are responsible for schistosomiasis (Figure \(\PageIndex{4}\)). Immature forms burrow through the skin into the blood. They migrate to the lungs, then to the liver and, later, other organs. Symptoms include anemia, malnutrition, fever, abdominal pain, fluid buildup, and sometimes death.
The other medically important group of platyhelminths are commonly known as tapeworms (cestodes) and are segmented flatworms that may have suckers or hooks at the scolex (head region) (Figure \(\PageIndex{3}\)). Tapeworms use these suckers or hooks to attach to the wall of the small intestine. The body of the worm is made up of segments called proglottid s that contain reproductive structures; these detach when the gametes are fertilized, releasing gravid proglottids with eggs. Tapeworms often have an intermediate host that consumes the eggs, which then hatch into a larval form called an oncosphere. The oncosphere migrates to a particular tissue or organ in the intermediate host, where it forms cysticerci. After being eaten by the definitive host, the cysticerci develop into adult tapeworms in the host's digestive system (Figure \(\PageIndex{5}\)). Taenia saginata (the beef tapeworm) and T. solium (the pork tapeworm) enter humans through ingestion of undercooked, contaminated meat. The adult worms develop and reside in the intestine, but the larval stage may migrate and be found in other body locations such as skeletal and smooth muscle. The beef tapeworm is relatively benign, although it can cause digestive problems and, occasionally, allergic reactions. The pork tapeworm can cause more serious problems when the larvae leave the intestine and colonize other tissues, including those of the central nervous system. Diphylobothrium latum is the largest human tapeworm and can be ingested in undercooked fish. It can grow to a length of 15 meters. Echinococcus granulosus , the dog tapeworm, can parasitize humans and uses dogs as an important host.
What group of medically important flatworms is segmented and what group is unsegmented?
For residents of temperate, developed countries, it may be difficult to imagine just how common helminth infections are in the human population. In fact, they are quite common and even occur frequently in the United States. Worldwide, approximately 807–1,221 million people are infected with Ascaris lumbricoides (perhaps one-sixth of the human population) and far more are infected if all nematode species are considered. 2 Rates of infection are relatively high even in industrialized nations. Approximately 604–795 million people are infected with whipworm ( Trichuris ) worldwide ( Trichuris can also infect dogs), and 576–740 million people are infected with hookworm ( Necator americanus and Ancylostoma duodenale ). 3 Toxocara, a nematode parasite of dogs and cats, is also able to infect humans. It is widespread in the United States, with about 10,000 symptomatic cases annually. However, one study found 14% of the population (more than 40 million Americans) was seropositive, meaning they had been exposed to the parasite at one time. More than 200 million people have schistosomiasis worldwide. Most of the World Health Organization (WHO) neglected tropical diseases are helminths. In some cases, helminths may cause subclinical illnesses, meaning the symptoms are so mild that that they go unnoticed. In other cases, the effects may be more severe or chronic, leading to fluid accumulation and organ damage. With so many people affected, these parasites constitute a major global public health concern.
Dracunculiasis, or Guinea worm disease, is caused by a nematode called Dracunculus medinensis . When people consume contaminated water, water fleas (small crustaceans) containing the nematode larvae may be ingested. These larvae migrate out of the intestine, mate, and move through the body until females eventually emerge (generally through the feet). While Guinea worm disease is rarely fatal, it is extremely painful and can be accompanied by secondary infections and edema (Figure \(\PageIndex{6}\)).
An eradication campaign led by WHO, the CDC, the United Nations Children’s Fund (UNICEF), and the Carter Center (founded by former U.S. president Jimmy Carter) has been extremely successful in reducing cases of dracunculiasis. This has been possible because diagnosis is straightforward, there is an inexpensive method of control, there is no animal reservoir, the water fleas are not airborne (they are restricted to still water), the disease is geographically limited, and there has been a commitment from the governments involved. Additionally, no vaccines or medication are required for treatment and prevention. In 1986, 3.5 million people were estimated to be affected. After the eradication campaign, which included helping people in affected areas learn to filter water with cloth, only four countries continue to report the disease (Chad, Mali, South Sudan, and Ethiopia) with a total of 126 cases reported to WHO in 2014.
Key Concepts and Summary
- Helminth parasites are included within the study of microbiology because they are often identified by looking for microscopic eggs and larvae.
- The two major groups of helminth parasites are the roundworms (Nematoda) and the flatworms (Platyhelminthes).
- Nematodes are common intestinal parasites often transmitted through undercooked foods, although they are also found in other environments.
- Platyhelminths include tapeworms and flukes , which are often transmitted through undercooked meat.
Footnotes
- 1 Won K, Kruszon-Moran D, Schantz P, Jones J. “National seroprevalence and risk factors for zoonotic Toxocara spp. infection.” In: Abstracts of the 56th American Society of Tropical Medicine and Hygiene; Philadelphia, Pennsylvania; 2007 Nov 4-8.
- 2 Fenwick, A. “The global burden of neglected tropical diseases.” Public health 126 no.3 (Mar 2012): 233–6.
- 3 de Silva, N., et. al. (2003). “Soil-transmitted helminth infections: updating the global picture”. Trends in Parasitology 19 (December 2003): 547–51.
- 4 World Health Organization. “South Sudan Reports Zero Cases of Guinea-Worm Disease for Seventh Consecutive Month.” 2016. http://www.who.int/dracunculiasis/no...ive_months/en/ . Accessed May 2, 2016.


4.1: An introduction to Eigenvalues and Eigenvectors
\(\newcommand{\twovec}[2]{\begin{pmatrix} #1 \\ #2 \end{pmatrix} } \) \(\newcommand{\threevec}[3]{\begin{pmatrix} #1 \\ #2 \\ #3 \end{pmatrix} } \) \(\newcommand{\fourvec}[4]{\begin{pmatrix} #1 \\ #2 \\ #3 \\ #4 \end{pmatrix} } \) \(\newcommand{\fivevec}[5]{\begin{pmatrix} #1 \\ #2 \\ #3 \\ #4 \\ #5 \end{pmatrix} } \)
This section introduces the concept of eigenvalues and eigenvectors and offers an example that motivates our interest in them. The point here is to develop an intuitive understanding of eigenvalues and eigenvectors and explain how they can be used to simplify some problems that we have previously encountered. In the rest of this chapter, we will develop this concept into a richer theory and illustrate its use with more meaningful examples.
Preview Activity 4.1.1.
Before we introduce the definition of eigenvectors and eigenvalues, it will be helpful to remember some ideas we have seen previously.
-
Suppose that \(\mathbf v\) is the vector shown in the figure. Sketch the vector \(2\mathbf v\) and the vector \(-\mathbf v\text{.}\)
- State the geometric effect that scalar multiplication has on the vector \(\mathbf v\text{.}\) Then sketch all the vectors of the form \(\lambda \mathbf v\) where \(\lambda\) is a scalar.
-
State the geometric effect of the matrix transformation defined by
\begin{equation*} \left[\begin{array}{rr} 3 & 0 \\ 0 & -1 \\ \end{array}\right]\text{.} \end{equation*}
-
Suppose that \(A\) is a \(2\times 2\) matrix and that \(\mathbf v_1\) and \(\mathbf v_2\) are vectors such that
\begin{equation*} A\mathbf v_1 = 3 \mathbf v_1, \qquad A\mathbf v_2 = -\mathbf v_2\text{.} \end{equation*}
Use the linearity of matrix multiplication to express the following vectors in terms of \(\mathbf v_1\) and \(\mathbf v_2\text{.}\)
- \(A(4\mathbf v_1)\text{.}\)
- \(A(\mathbf v_1 + \mathbf v_2)\text{.}\)
- \(A(4\mathbf v_1 -3\mathbf v_2)\text{.}\)
- \(A^2\mathbf v_1\text{.}\)
- \(A^2(4\mathbf v_1 - 3\mathbf v_2)\text{.}\)
- \(A^4\mathbf v_1\text{.}\)
A few examples
We will now introduce the definition of eigenvalues and eigenvectors and then look at a few simple examples.
Given a square \(n\times n\) matrix \(A\text{,}\) we say that a nonzero vector \(\mathbf v\) is an eigenvector of \(A\) if there is a scalar \(\lambda\) such that
The scalar \(\lambda\) is called the eigenvalue associated to the eigenvector \(\mathbf v\text{.}\)
At first glance, there is a lot going on in this definition so let's look at an example.
Consider the matrix \(A = \left[\begin{array}{rr} 7 & 6 \\ 6 & -2 \\ \end{array}\right]\) and the vector \(\mathbf v=\twovec{2}{1}\text{.}\) We find that
In other words, \(A\mathbf v = 10\mathbf v\text{,}\) which says that \(\mathbf v\) is an eigenvector of the matrix \(A\) with associated eigenvalue \(\lambda = 10\text{.}\)
Similarly, if \(\mathbf w = \twovec{-1}{2}\text{,}\) we find that
Here again, we have \(A\mathbf w = -5\mathbf w\) showing that \(\mathbf w\) is an eigenvector of \(A\) with associated eigenvalue \(\lambda=-5\text{.}\)
Activity 4.1.2.
This definition has an important geometric interpretation that we will investigate here.
- Suppose that \(\mathbf v\) is a nonzero vector and that \(\lambda\) is a scalar. What is the geometric relationship between \(\mathbf v\) and \(\lambda\mathbf v\text{?}\)
- Let's now consider the eigenvector condition: \(A\mathbf v = \lambda\mathbf v\text{.}\) Here we have two vectors, \(\mathbf v\) and \(A\mathbf v\text{.}\) If \(A\mathbf v = \lambda\mathbf v\text{,}\) what is the geometric relationship between \(\mathbf v\) and \(A\mathbf v\text{?}\)
- The sliders in the diagram below allow you to choose a matrix \(A\). The vector \({\mathbf v}\) is shown in red and may be varied by clicking in the head of the vector. The vector \(A{\mathbf v}\) is then shown in grayChoose the matrix \(A= \left[\begin{array}{rr} 1& 2 \\ 2& 1 \\ \end{array}\right] \text{.}\) Move the vector \(\mathbf v\) so that the eigenvector condition holds. What is the eigenvector \(\ mathbf v\) and what is the associated eigenvalue?
- By algebraically computing \(A\mathbf v\text{,}\) verify that the eigenvector condition holds for the vector \(\mathbf v\) that you found.
- If you multiply the eigenvector \(\mathbf v\) that you found by \(2\text{,}\) do you still have an eigenvector? If so, what is the associated eigenvalue?
- Are you able to find another eigenvector \(\mathbf v\) that is not a scalar multiple of the first one that you found? If so, what is the eigenvector and what is the associated eigenvalue?
- Now consider the matrix \(A = \left[\begin{array}{rr} 2 & 1 \\ 0 & 2 \\ \end{array}\right] \text{.}\) Use the diagram to describe any eigenvectors and associated eigenvalues.
- Finally, consider the matrix \(A = \left[\begin{array}{rr} 0 & -1 \\ 1 & 0 \\ \end{array}\right] \text{.}\) Use the diagram to describe any eigenvectors and associated eigenvalues. What geometric transformation does this matrix perform on vectors? How does this explain the presence of any eigenvectors?
Let's consider the ideas we saw in the activity in some more depth. To be an eigenvector of \(A\text{,}\) the vector \(\mathbf v\) must satisfy \(A\mathbf v = \lambda\mathbf v\) for some scalar \(\lambda\text{.}\) This means that \(\mathbf v\) and \(A\mathbf v\) are scalar multiples of one another, which means they must lie on the same line.
Consider now the matrix \(A = \left[\begin{array}{rr} 1 & 2 \\ 2 & 1 \\ \end{array}\right] \text{.}\) On the left of Figure 4.1.3, we see that \(\mathbf v=\twovec{1}{0}\) is not an eigenvector of \(A\) since the vectors \(\mathbf v\) and \(A\mathbf v\) do not lie on the same line. On the right, however, we see that \(\mathbf v=\twovec{1}{1}\) is an eigenvector. In fact, \(A\mathbf v\) is obtained from \(\mathbf v\) by stretching \(\mathbf v\) by a factor of \(3\text{.}\) Therefore, \(\mathbf v\) is an eigenvector of \(A\) with eigenvalue \(\lambda = 3\text{.}\)
It is not difficult to see that any multiple of \(\twovec{1}{1}\) is also an eigenvector of \(A\) with eigenvalue \(\lambda = 3\text{.}\) Indeed, we will see later that all the eigenvectors associated to a given eigenvalue form a subspace of \(\mathbb R^n\text{.}\)
In Figure 4.1.4, we see that \(\mathbf v=\twovec{-1}{1}\) is also an eigenvector with eigenvalue \(\lambda =-1\text{.}\)
The interactive diagram we used in the activity is meant to convey the fact that the eigenvectors of a matrix \(A\) are special vectors. Most of the time, the vectors \(\mathbf v\) and \(A\mathbf v\) appear somewhat unrelated. For certain vectors, however, \(\mathbf v\) and \(A\mathbf v\) line up with one another. Something important is going on when that happens so we call attention to those vectors by calling them eigenvectors. For these vectors, the operation of multiplying by \(A\) reduces to the much simpler operation of scalar multiplying by \(\lambda\text{.}\) The reason eigenvectors are important is because it is extremely convenient to be able to replace matrix multiplication by scalar multiplication.
Eigen is a German word that can be interpreted as meaning “characteristic”. As we will see, the eigenvectors and eigenvalues of a matrix \(A\) give an important characterization of the matrix.
The usefulness of eigenvalues and eigenvectors
In the next section, we will introduce an algebraic technique for finding the eigenvalues and eigenvectors of a matrix. Before doing that, however, we would like to discuss why eigenvalues and eigenvectors are so useful.
Let's continue looking at the example \(A = \left[\begin{array}{rr} 1 & 2 \\ 2 & 1 \\ \end{array}\right] \text{.}\) We have seen that \(\mathbf v_1 = \twovec{1}{1}\) is an eigenvector with eigenvalue \(\lambda=3\) and \(\mathbf v_2=\twovec{-1}{1}\) is an eigenvector with eigenvalue \(\lambda = -1\text{.}\) This means that \(A\mathbf v_1 = 3\mathbf v_1\) and \(A\mathbf v_2=-\mathbf v_2\text{.}\) By the linearity of matrix multiplication, we can easily understand what happens when we multiply a linear combination of \(\mathbf v_1\) and \(\mathbf v_2\) by \(A\text{:}\)
For instance, if we consider the vector \(\mathbf x=\mathbf v_1-2\mathbf v_2\text{,}\) we find that
as seen in the figure.
In other words, multiplying by \(A\) has the effect of stretching a vector \(\mathbf x\) in the \(\mathbf v_1\) direction by a factor of \(3\) and flipping the \(\mathbf x\) in \(\mathbf v_2\) direction.
We can draw an analogy with the more familiar example of the diagonal matrix \(D=\left[\begin{array}{rr} 3 & 0 \\ 0 & -1 \\ \end{array}\right] \text{.}\) As we have seen, the effect of multiplying a vector \(\mathbf x\) by \(D\) is to stretch \(\mathbf x\) horizontally by a factor of \(3\) and to flip \(\mathbf x\) vertically. This is illustrated in Figure 4.1.5.
The matrix \(A=\left[\begin{array}{rr} 1 & 2 \\ 2 & 1 \\ \end{array}\right]\) has a similar effect when viewed in the basis defined by the eigenvectors \(\mathbf v_1\) and \(\mathbf v_2\text{,}\) as seen in Figure 4.1.6.
In a sense that will be made precise later, having a set of eigenvectors of \(A\) that forms a basis of \(\mathbb R^2\) enables us to think of \(A\) as being equivalent to a diagonal matrix \(D\text{.}\) Of course, as the other examples in the previous activity show, it may not always be possible to form a basis from the eigenvectors of a matrix. For example, the only eigenvectors of the matrix \(\left[\begin{array}{rr} 2 & 1 \\ 0 & 2 \end{array}\right]\text{,}\) which represents a shear, have the form \(\twovec{x}{0}\text{.}\) In this example, we are not able to create a basis for \(\mathbb R^2\) consisting of eigenvectors of the matrix. This is also true for the matrix \(\left[\begin{array}{rr} 0 & -1 \\ 1 & 0 \\ \end{array}\right] \text{,}\) which represents a \(90^\circ\) rotation.
Activity 4.1.3.
Let's consider an example that illustrates how we can put these ideas to use.
Suppose that we work for a car rental company that has two locations, \(P\) and \(Q\text{.}\) When a customer rents a car at one location, they have the option to return it to either location at the end of the day. After doing some market research, we determine:
- 80% of the cars rented at location \(P\) are returned to \(P\) and 20% are returned to \(Q\text{.}\)
- 40% of the cars rented at location \(Q\) are returned to \(Q\) and 60% are returned to \(P\text{.}\)
- Suppose that there are 1000 cars at location \(P\) and no cars at location \(Q\) on Monday morning. How many cars are there are locations \(P\) and \(Q\) at the end of the day on Monday?
- How many are at locations \(P\) and \(Q\) at end of the day on Tuesday?
-
If we let \(P_k\) and \(Q_k\) be the number of cars at locations \(P\) and \(Q\text{,}\) respectively, at the end of day \(k\text{,}\) we then have
\begin{equation*} \begin{aligned} P_{k+1}& {}={} 0.8P_k + 0.6Q_k \\ Q_{k+1}& {}={} 0.2P_k + 0.4Q_k\text{.} \\ \end{aligned} \end{equation*}
We can write the vector \(\mathbf x_k = \twovec{P_k}{Q_k}\) to reflect the number of cars at the two locations at the end of day \(k\text{,}\) which says that
\begin{equation*} \begin{aligned} \mathbf x_{k+1} & {}={} \left[\begin{array}{rr} 0.8 & 0.6 \\ 0.2 & 0.4 \\ \end{array}\right] \mathbf x_k \\ \\ \text{or}\qquad \mathbf x_{k+1} & {}={} A\mathbf x_k \end{aligned} \end{equation*}where \(A=\left[\begin{array}{rr}0.8 & 0.6 \\ 0.2 & 0.4 \end{array}\right]\text{.}\)
Suppose that
\begin{equation*} \mathbf v_1 = \twovec{3}{1}, \qquad \mathbf v_2 = \twovec{-1}{1}\text{.} \end{equation*}Compute \(A\mathbf v_1\) and \(A\mathbf v_2\) to demonstrate that \(\mathbf v_1\) and \(\mathbf v_2\) are eigenvectors of \(A\text{.}\) What are the associated eigenvalues \(\lambda_1\) and \(\lambda_2\text{?}\)
- We said that 1000 cars are initially at location \(P\) and none at location \(Q\text{.}\) This means that the initial vector describing the number of cars is \(\mathbf x_0 = \twovec{1000}{0}\text{.}\) Write \(\mathbf x_0\) as a linear combination of \(\mathbf v_1\) and \(\mathbf v_2\text{.}\)
- Remember that \(\mathbf v_1\) and \(\mathbf v_2\) are eigenvectors of \(A\text{.}\) Use the linearity of matrix multiplicaiton to write the vector \(\mathbf x_1 = A\mathbf x_0\text{,}\) describing the number of cars at the two locations at the end of the first day, as a linear combination of \(\mathbf v_1\) and \(\mathbf v_2\text{.}\)
-
Write the vector \(\mathbf x_2 = A\mathbf x_1\) as a linear combination of \(\mathbf v_1\) and \(\mathbf v_2\text{.}\) Then write the next few vectors as linear combinations of \(\mathbf v_1\) and \(\mathbf v_2\text{:}\)
- \(\mathbf x_3 = A\mathbf x_2\text{.}\)
- \(\mathbf x_4 = A\mathbf x_3\text{.}\)
- \(\mathbf x_5 = A\mathbf x_4\text{.}\)
- \(\mathbf x_6 = A\mathbf x_5\text{.}\)
- What will happen to the number of cars at the two locations after a very long time? Explain how writing \(\mathbf x_0\) as a linear combination of eigenvectors helps you determine the long-term behavior.
This activity is important and motivates much of our work with eigenvalues and eigenvectors so we will review it now making sure we have a clear understanding of the concepts that arise.
First, we compute
This shows that \(\mathbf v_1\) is an eigenvector of \(A\) with eigenvalue \(\lambda_1 = 1\) and \(\mathbf v_2\) is an eigenvector of \(A\) with eigenvalue \(\lambda_2=0.2\text{.}\)
By the linearity of matrix matrix multiplication, we have
Therefore, we will write the vector describing the initial distribution of cars \(\mathbf x_0=\twovec{1000}{0}\) as a linear combination of \(\mathbf v_1\) and \(\mathbf v_2\text{;}\) that is, \(\mathbf x_0 = c_1\mathbf v_2 + c_2 \mathbf v_2\text{.}\) To do, we form the augmented matrix and row reduce:
Therefore, \(\mathbf x_0 = 250\mathbf v_1 -250\mathbf v_2\text{.}\)
To determine the distribution of cars on subsequent days, we will repeatedly multiply by \(A\text{.}\) We find that
In particular, this shows us that
Taking notice of the pattern, we may write
Multiplying a number by \(0.2\) is the same as taking 20% of that number. As each day goes by, the second term is multiplied by \(0.2\) so the coefficient of \(\mathbf v_2\) in the expression for \(\mathbf x_k\) will eventually become extremely small. We therefore see that the distribution of cars will stabilize at \(\mathbf x=250\mathbf v_1=\twovec{750}{250}\text{.}\)
Notice how our use of the eigenvalues and eigenvectors of \(A\) enable us to look far into the future without having to repeatedly multiply a vector by the matrix \(A\text{.}\) Knowing the eigenvectors allows us to replace matrix multiplication with the simpler operation of scalar multiplication. This is a powerful tool that we will develop more in the rest of this chapter.
Notice also how this example relies on the fact that we can express the initial vector \(\mathbf x_0\) as a linear combination of eigenvectors. For this reason, we would like, given an \(n\times n\) matrix, to be able to create a basis of \(\mathbb R^n\) of eigenvectors. We will frequently return to this question in later sections.
Question 4.1.7.
Given an \(n\times n\) matrix \(A\text{,}\) when can we form a basis for \(\mathbb R^n\) consisting of eigenvectors of \(A\text{?}\)
Summary
We defined an eigenvector of a square matrix \(A\) to be a nonzero vector \(\mathbf v\) such that \(A\mathbf v = \lambda\mathbf v\) for some scalar \(\lambda\text{,}\) which is called the eigenvalue associated to \(\mathbf v\text{.}\)
- If \(\mathbf v\) is an eigenvector, then matrix multiplication by \(A\) reduces to the simpler operation of scalar multiplication by \(\lambda\text{.}\)
- Scalar multiples of an eigenvector are also eigenvectors. In fact, we will see that the eigenvectors associated to an eigenvalue \(\lambda\) form a subspace.
- If we can form a basis for \(\mathbb R^n\) consisting of eigenvectors of \(A\text{,}\) then \(A\) is, in some sense, equivalent to a diagonal matrix.
- Rewriting a vector \(\mathbf x\) as a linear combination of eigenvectors of \(A\) makes it easy to repeatedly multiply \(\mathbf x\) by \(A\text{.}\)
Exercises 4.1.4Exercises
Consider the matrix and vectors
-
Show that the vectors \(\mathbf v_1\) and \(\mathbf v_2\) are eigenvectors of \(A\) and find their associated eigenvalues.
- Express the vector \(\mathbf x = \twovec{-4}{-1}\) as a linear combination of \(\mathbf v_1\) and \(\mathbf v_2\text{.}\)
- Use this expression to compute \(A\mathbf x\text{,}\) \(A^2\mathbf x\text{,}\) and \(A^{-1}\mathbf x\) as a linear combination of eigenvectors.
Consider the matrix and vectors
-
Show that the vectors \(\mathbf v_1\text{,}\) \(\mathbf v_2\text{,}\) and \(\mathbf v_3\) are eigenvectors of \(A\) and find their associated eigenvalues.
- Express the vector \(\mathbf x = \threevec{0}{-3}{-4}\) as a linear combination of the eigenvectors.
- Use this expression to compute \(A\mathbf x\text{,}\) \(A^2\mathbf x\text{,}\) and \(A^{-1}\mathbf x\) as a linear combination of eigenvectors.
Suppose that \(A\) is an \(n\times n\) matrix.
- Explain why \(\lambda = 0\) is an eigenvalue if and only if there is a non-trivial solution to the homogeneous equation \(A\mathbf x = 0\text{.}\)
- Explain why such a matrix \(A\) is not invertible if and only if \(\lambda=0\) is an eigenvalue.
- If \(A\) is an invertible \(n\times n\) matrix having an eigenvector \(\mathbf v\) and associated eigenvalue \(\lambda\text{,}\) explain why \(\mathbf v\) is also an eigenvector of \(A^{-1}\) with associated eigenvalue \(\lambda^{-1}\text{.}\)
- If \(A\) is an \(n\times n\) matrix with eigenvector \(\mathbf v\) and associated eigenvalue \(\lambda\text{,}\) explain why \(\mathbf v\) is also an eigenvector of \(A^2\) with associated eigenvalue \(\lambda^2\text{.}\)
- The matrix \(A=\ \left[\begin{array}{rr} 1 & 2 \\ 2 & 1 \\ \end{array}\right]\) has eigenvectors \(\mathbf v_1=\twovec{1}{1}\) and \(\mathbf v_2=\twovec{-1}{1}\) and associated eigenvalues \(\lambda_1 = 3\) and \(\lambda=-1\text{.}\) What are some eigenvectors and associated eigenvalues for \(A^5\text{?}\)
Suppose that \(A\) is a matrix with eigenvectors \(\mathbf v_1\) and \(\mathbf v_2\) and eigenvalues \(\lambda_1 = -1\) and \(\lambda_2=2\) as shown.
Sketch the vectors \(A\mathbf x\text{,}\) \(A^2\mathbf x\text{,}\) and \(A^{-1}\mathbf x\text{.}\)
For the following matrices, find the eigenvectors and associated eigenvalues by thinking geometrically about the corresponding matrix transformation.
- \(\left[\begin{array}{rr} 3 & 0 \\ 0 & 3 \\ \end{array}\right] \text{.}\)
- \(\left[\begin{array}{rr} -2 & 0 \\ 0 & 4 \\ \end{array}\right] \text{.}\)
- What are the eigenvectors and associated eigenvalues of the identity matrix?
- What are the eigenvectors and associated eigenvalues of a diagonal matrix with distinct diagonal entries?
Suppose that \(A\) is a \(2\times2\) matrix having eigenvectors
and associated eigenvalues \(\lambda_1=2\) and \(\lambda_2=-3\text{.}\) If \(\mathbf x=\twovec{5}{0}\text{,}\) find the vector \(A^4\mathbf x\text{.}\)
Determine whether the following statements are true or false and provide a justification for your response.
- The eigenvalues of a diagonal matrix are equal to the entries on the diagonal.
- If \(A\mathbf v=\lambda\mathbf v\text{,}\) then \(A^2\mathbf v=\lambda\mathbf v\) as well.
- Every vector is an eigenvector of the identity matrix.
- If \(\lambda=0\) is an eigenvalue of \(A\text{,}\) then \(A\) is invertible.
- For every \(n\times n\) matrix \(A\text{,}\) it is possible to find a basis of \(\mathbb R^n\) consisting of eigenvectors of \(A\text{.}\)
Consider the two special matrices below and find their eigenvectors and associated eigenvalues.
- \(\left[\begin{array}{rrr} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{array}\right] \text{.}\)
- \(\left[\begin{array}{rrr} 1 & 1 & 1 \\ 2 & 2 & 2 \\ 3 & 3 & 3 \\ \end{array}\right] \text{.}\)
For each of the following matrix transformations, describe the eigenvalues and eigenvectors of the corresponding matrix \(A\text{.}\)
- A reflection in \(\mathbb R^2\) in the line \(y=x\text{.}\)
- A \(180^\circ\) rotation in \(\mathbb R^2\text{.}\)
- A \(180^\circ\) rotation in \(\mathbb R^3\) about the \(y\)-axis.
- A \(90^\circ\) rotation in \(\mathbb R^3\) about the \(x\)-axis.
Suppose we have two species, \(P\) and \(Q\text{,}\) where species \(P\) preys on \(Q\text{.}\) Their populations, in millions, in year \(k\) are denoted by \(P_k\) and \(Q_k\) and satisfy
We will keep track of the populations in year \(k\) using the vector \(\mathbf x_k=\twovec{P_k}{Q_k}\) so that
- Show that \(\mathbf v_1=\twovec{1}{3}\) and \(\mathbf v_2=\twovec{2}{1}\) are eigenvectors of \(A\) and find their associated eigenvalues.
- Suppose that the initial populations are described by the vector \(\mathbf x_0 = \twovec{38}{44}\text{.}\) Express \(\mathbf x_0\) as a linear combination of \(\mathbf v_1\) and \(\mathbf v_2\text{.}\)
- Find the populations after one year, two years, and three years by writing the vectors \(\mathbf x_1\text{,}\) \(\mathbf x_2\text{,}\) and \(\mathbf x_3\) as linear combinations of \(\mathbf v_1\) and \(\mathbf v_2\text{.}\)
- What is the general form for \(\mathbf x_k\text{?}\)
- After a very long time, what is the ratio of \(P_k\) to \(Q_k\text{?}\)


10.8: Review Questions
What is pivoting?
2 .How is pitching different in the lean startup method?
3 .What is the lean startup process?
4 .What are some examples of an MVP?
5 .What is the success rate of businesses in the first two to ten years of life in the United States?
6 .How can an entrepreneur learn from failures?
7 .Which gender experiences more fear about opening a business? Why do you think it they are more fearful?
8 .What are some of the realities of business ownership?
9 .How can entrepreneurs deal with these negative realities?
10 .What percentage of businesses last five years or more?
11 .What percentage of US businesses are minority owned?
12 .What is a lean plan and what are its components?
13 .What are some of the items or sections that do not have to be included in a lean plan?
14 .What kind of lifecycles do businesses go through? Why?
15 .How does each phase affect business decisions?


2.5: Positive Visualization
Henry Ford once said, “Whether you think you can, or you think you can’t, you’re right.” What does a top-level basketball player do to prepare to take a free-throw shot? The player often goes through a routine, usually by dribbling a certain number of times, then assumes the shooting stance, and pauses. At this point, athletes like this, as well as successful professionals and public speakers, employ a technique called positive visualization or positive imagery , in which they pause a moment to imagine the best possible outcome before taking action. That basketball player, in the brief moment before taking the shot, imagines the ball arcing perfectly through the air, and swishing through the hoop, touching nothing but net. Consider these real-life examples:
| Muhammad Ali always stressed the importance of imagining himself victorious long before his actual fight. | |
| Michael Jordan would always take the last shot in his mind before he took that shot in reality. | |
| As a young actor, Jim Carrey said he pictured himself being the greatest actor in the world. |
So why does this technique seem to work? Can some mystical psychic explain this phenomenon? Once again, biology holds the key to understanding the benefit of this practice technique. The human imagination is a powerful tool. CT scans taken of the human brain during exercises involving the imagination demonstrate that the very same areas of the brain that activate during a real-life activity also activate when a person imagines performing the same activity (Schlegel, Kohler, Fogelson, Alexander, Konuthula, & Ulric Tse, 2013). This means people can actually train their brains to perform effectively by employing positive visualization and imagining successful outcomes. When combined with practicing a speech as close as possible to the desired end result, this technique works powerfully to reduce anxiety, eliminate uncertainties, and increase self-confidence.


9.6: Modeling more than one Discrete Character at a Time
It is extremely common to have datasets with more than one discrete character – in fact, one could argue that multivariate discrete datasets are the cornerstone of systematics. Nowadays, the most common multivariate discrete datasets are composed of genetic/genomic data. However, the foundations of modern phylogenetic comparative biology were laid out by Hennig (1966) and the other early cladists, who worked out methods for using discrete character data to obtain phylogenetic trees that show the evolutionary history of clades.
Almost all phylogenetic reconstruction methods that use discrete characters as data make a key assumption: that each of these characters evolves independently from one another. Mathematically, one calculates the likelihood for each single character, then multiplies this likelihood (or, equivalently, adds the log-likelihood) across all characters to obtain the likelihood of the data.
The assumption of character independence is clearly not true in general. In the case of morphological characters, structures often interact with one another to determine the fitness of an individual, and it seems very likely that those structures are not independent. In fact, some times we are specifically interested in whether or not particular sets of characters evolve independently or not. Methods that assume character independence a priori are not useful for that sort of framework.
Felsenstein (1985) made a huge impact on the field of evolutionary biology with a statistical argument about species: species can not be considered independent data points because they share an evolutionary history. Species that are most closely related to one another will covary, simply due to that shared history. Nowadays, one cannot publish a paper in comparative biology without accounting directly for the non-independence of species that evolve on a tree. However, it is still very common to ignore the non-independence of characters, even when they occur together in the same organism! Surely the shared developmental history of two characters within one body commonly leads to correlations across these characters.