An Inquiry-Based Introduction to Linear Algebra and Applications

Subsection Application: The Bailey Model of an Epidemic

The COVID-19 epidemic has generated many mathematical and statistical models to try to understand the spread of the virus. In 1950 Norman Bailey proposed a simple stochastic model of the spread of an epidemic. The solution to the model involves matrix exponentials and the Jordan canonical form is a useful tool for calculating matrix exponentials.

Subsection Introduction

We have seen several different matrix factorizations so far, eigenvalue decomposition (Section 19), singular value decomposition (Section 29), QR factorization (Section 25), and LU factorization (Section 22). In this section, we investigate the Jordan canonical form, which in a way generalizes the eigenvalue decomposition. For matrices with an eigenvalue decomposition, the geometric multiplicity of each eigenvalue (the dimension of the corresponding eigenspace) must equal its algebraic multiplicity (the number of times the eigenvalue occurs as a root of the characteristic polynomial of the matrix). We know that not every matrix has an eigenvalue decomposition. However, every square matrix has a Jordan canonical form, in which we use generalized eigenvectors and block diagonal form to approximate the eigenvalue decomposition behavior. At the end of the section we provide a complete proof of the existence of the Jordan canonical form.

Subsection When an Eigenvalue Decomposition Does Not Exist

Recall that an eigenvalue decomposition of a matrix exists if and only if the algebraic multiplicity equals the geometric multiplicity for each eigenvalue. In this case, the whole space \(\R^n\) decomposes into eigenspaces corresponding to the eigenvalues and the matrix acts as scalar multiplication in each eigenspace. We consider some cases where such a decomposition exists and some where it does not to notice some differences between the two cases and think of ways to improvise the situation.

Subsection Generalized Eigenvectors and the Jordan Canonical Form

If an \(n \times n\) matrix \(A\) has \(n\) linearly independent eigenvectors, then \(A\) is similar to a diagonal matrix with the eigenvalues along the diagonal. You discovered in Preview Activity 40.1 a matrix without enough linearly independent eigenvectors can look close enough to a diagonal matrix. We now investigate whether this generalized representation can be achieved for all matrices.

Activity 40.2 shows that even if a matrix does not have a full complement of linearly independent eigenvectors, we can still almost diagonalize the matrix. If \(A\) is an \(n \times n\) matrix that does not have \(n\) linearly independent eigenvectors (such matrices area called defective), the matrix \(A\) is still similar to a matrix that is “close” to diagonal. Activity 40.2 shows this for the case of a \(3 \times 3\) matrix \(M\) with two eigenvalues, \(\lambda_1, \lambda_2\) with \(\lambda_2\) having algebraic multiplicity two, and each eigenvalue with one-dimensional eigenspace. Let \(\vv_1\) be an eigenvector corresponding to \(\lambda_1\text{.}\) Because \(\lambda_2\) also has a one-dimensional eigenspace, the matrix \(M\) has only two linearly independent eigenvectors. Nevertheless, as you saw in Activity 40.2, we can find a vector \(\vv_3\) that is in \(\Nul (M - \lambda_2 I)^2\) but not in \(\Nul (M-\lambda_2 I)\text{.}\) From this, it follows that the vector \(\vv_2 = (M - \lambda_2 I) \vv_3\) is an eigenvector of \(M\) with eigenvalue \(\lambda_2\text{.}\) In this case we have \(M \vv_3 = \lambda_2 \vv_3 + \vv_2\) and

The lack of two linearly independent eigenvectors for this eigenvalue of algebraic multiplicity two will ensure that \(M\) is not similar to a diagonal matrix, but \(M\) is similar to a matrix with a diagonal block of the form \(\left[ \begin{array}{cc} \lambda \amp 1 \\ 0 \amp \lambda \end{array} \right]\text{.}\) So even though the matrix \(M\) is not diagonalizable, we can find an invertible matrix \(C = [\vv_1 \ \vv_2 \ \vv_3]\) so that \(C^{-1}AC\) is almost diagonalizable.

In general, suppose we have an eigenvalue \(\lambda\) of a matrix \(A\) with algebraic multiplicity two but with a one-dimensional eigenspace. Then the eigenvalue \(\lambda\) is deficient — that is \(\dim(E_{\lambda})\) is strictly less than the algebraic multiplicity of \(\lambda\text{,}\) i.e. \(E_{\lambda}\) does not contain enough linearly independent eigenvectors for \(\lambda\text{.}\) But as we saw above, we may be able to find a vector \(\vv_2\) that is in \(\Nul (A - \lambda I)^2\) but not in \(\Nul (A - \lambda I)\text{.}\) When we let \(\vv_1 = (A-\lambda I) \vv_2\text{,}\) we then also have

as we argued in Preview Activity 40.1, and so \(\vv_1\) is an eigenvector for \(A\) with eigenvalue \(\lambda\text{.}\) It is vectors that satisfy an equation like (40.1) that drive the Jordan canonical form. These vectors are similar to eigenvectors and are called generalized eigenvectors.

In other words, a generalized eigenvector of an \(n \times n\) matrix \(A\) corresponding to the eigenvalue \(\lambda\) is a nonzero vector in \(\Nul (A - \lambda I_n)^m\) for some \(m\text{.}\) Note that every eigenvector of \(A\) is a generalized eigenvector (with \(m=1\)). In Preview Activity 40.1, \(A\) is a \(3 \times 3\) matrix with eigenvalue \(\lambda = 5\) having algebraic multiplicity 2 and geometric multiplicity 1. We were able to see that \(A\) is similar to a matrix of the form \(\left[ \begin{array}{ccc} 1 \amp 0\amp 0 \\ 0 \amp 5 \amp 1 \\ 0\amp 0\amp 5 \end{array} \right]\) because of the existence of a generalized eigenvector for the eigenvalue \(5\text{.}\)

The example in Preview Activity 40.1 presents the basic idea behind how we can find a “simple” matrix that is similar to any square matrix, even if that matrix is not diagonalizable. The key is to find generalized eigenvectors for eigenvalues whose algebraic multiplicities exceed their geometric multiplicities. One way to do this is indicated in Preview Activity 40.1 and in the next activity.

It is the equations \((A- 2I)\vv_{i+1} = \vv_i\) from Activity 40.3 that give us this simple form \(\left[ \begin{array}{ccc} 2\amp 1\amp 0 \\ 0\amp 2\amp 1 \\ 0\amp 0\amp 2 \end{array} \right]\text{.}\) To better understand why, notice that the equations imply that \(A\vv_{i+1} = 2\vv_{i+1} + \vv_i\text{.}\) So if \(C = [\vv_1 \ \vv_2 \ \vv_3]\text{,}\) then

This method will provide us with the Jordan canonical form. The major reason that this method always works is contained in the following theorem whose proof follows from the proof of the existence of the Jordan canonical form (presented later).

To find generalized eigenvectors, then, we find a value of \(p\) so that \(\dim(\Nul(A-\lambda I_n)^p) = m\) and then find a vector \(\vv_p\) that is in \(\Nul(A-\lambda I_n)^p\) but not in \(\Nul(A-\lambda I_n)^{p-1}\text{.}\) Successive multiplications by \(A - \lambda I_n\) provide a sequence of generalized eigenvectors. The sequence

is called a generalized eigenvector chain.

The previous activities illustrate the general idea for almost diagonalizing an arbitrary square matrix. First let \(A\) be an \(n \times n\) matrix with an eigenvalue \(\lambda\) of of algebraic multiplicity \(n\) and geometric multiplicity \(k\text{.}\) If \(\vv_1\text{,}\) \(\vv_2\text{,}\) \(\ldots\text{,}\) \(\vv_k\) are linearly independent eigenvectors for \(A\text{,}\) then we can extend the set as we did in Activity 40.4 above with generalized eigenvectors to a basis \(\{\vv_1, \vv_2, \ldots, \vv_k, \vv_{k+1}, \ldots, \vv_n\}\) of \(\R^n\text{.}\) The matrix \(C = [\vv_1 \ \vv_2 \ \cdots \ \vv_n]\) has the property that \(C^{-1}AC\) is almost diagonal. By almost, we mean that \(C^{_1}AC\) has block matrices along the diagonal that look like

and has zeros everywhere else. A matrix of the form (40.2) is called a Jordan block.

If \(A\) is an \(n \times n\) matrix with eigenvalues \(\lambda_1\text{,}\) \(\lambda_2\text{,}\) \(\ldots\text{,}\) \(\lambda_k\text{,}\) we repeat this process with every eigenvalue of \(A\) to construct an invertible matrix \(C\) so that \(C^{-1}AC\) is of the form

where each matrix \(J_i\) is a Jordan block (note that a \(1 \times 1\) Jordan block is allowable). The form in (40.3) is called the Jordan canonical form or Jordan normal form of the matrix \(A\text{.}\) Later in this section we will prove the following theorem.

Another example may help illustrate the process.

The previous activities highlight some of the information that a Jordan canonical tells us about a matrix. Assuming that \(C^{-1}AC = J\text{,}\) where \(J\) is in Jordan canonical form, we can say the following.

• Since similar matrices have the same eigenvalues, the eigenvalues of \(J\text{,}\) and therefore of \(A\text{,}\) are the diagonal entries of \(J\text{.}\) Moreover, the number of times a diagonal entry appears in \(J\) is the algebraic multiplicity of the eigenvalue. This is also the sum of the sizes of all Jordan blocks corresponding to \(\lambda\text{.}\)

• Given an eigenvalue \(\lambda\text{,}\) its geometric multiplicity is the number of Jordan blocks corresponding to \(\lambda\text{.}\)

• Each generalized eigenvector leads to a Jordan block for that eigenvector. The number of Jordan blocks corresponding to \(\lambda\) of size at least \(j\) is \(s_j = \dim\left(\Nul (A - \lambda I)^j\right) - \dim\left(\Nul (A - \lambda I)^{j-1}\right)\text{.}\) Thus, the number of Jordan blocks of size exactly \(j\) is

  \begin{equation*} s_j-s_{j+1} = 2 \dim\left(\Nul (A - \lambda I )^j\right) - \dim\left(\Nul (A - \lambda I )^{j+1}\right) - \dim\left(\Nul (A - \lambda I)^{j-1}\right)\text{.} \end{equation*}

One interesting consequence of the existence of the Jordan canonical form is the famous Cayley-Hamilton Theorem.

The proof of the Cayley-Hamilton Theorem follows from Exercise 22 that shows that every upper triangular matrix satisfies its characteristic polynomial. If \(A\) is a square matrix, then there exists a matrix \(C\) such that \(C^{-1}AC = T\text{,}\) where \(T\) is in Jordan canonical form (that is, \(T\) is upper triangular). So \(A\) is similar to \(T\text{.}\) If \(p(x)\) is the characteristic polynomial of \(A\text{,}\) Activity 19.5 in Section 19 tells us that \(p(x)\) is the characteristic polynomial of \(T\text{.}\) Therefore, \(p(T) = 0\text{.}\) Then Exercise 14 in Section 19 shows that \(p(A) = Cp(T)C^{-1} = 0\) and \(A\) satisfies its characteristic polynomial.

Subsection Geometry of Matrix Transformations using the Jordan Canonical Form

Recall that we can visualize the action of a matrix transformation defined by a diagonalizable matrix by using a change of basis. For example, let \(T(\vx) = A\vx\text{,}\) where \(A = \frac{1}{3}\left[ \begin{array}{rr} 8\amp -1\\-2\amp 7 \end{array} \right]\text{.}\) The eigenvalues of \(A\) are \(\lambda_1 = 3\) and \(\lambda_2 = 2\) with corresponding eigenvectors \(\vv_1 = [-1 \ 1]^{\tr}\) and \([1 \ 2]^{\tr}\text{.}\) So \(A\) is diagonalizable by the matrix \(P = \left[ \begin{array}{rc} -1\amp 1\\1\amp 2 \end{array} \right]\text{,}\) with \(P^{-1}AP = D = \left[ \begin{array}{cc} 3\amp 0\\0\amp 2 \end{array} \right]\text{.}\) Note that \(T(\vx) = PDP^{-1} \vx\text{.}\) Now \(P^{-1}\) is a change of basis matrix from the standard basis to the basis \(\CB = \{\vv_1, \vv_2\}\text{,}\) and \(D\) stretches space in the direction of \(\vv_1\) by a factor of 3 and stretches space in the direction of \(\vv_2\) by a factor of 2, and then \(P\) changes basis back to the standard basis. This is illustrated in Figure 40.5.

In general, if an \(n \times n\) matrix \(A\) is diagonalizable, then there is a basis \(\CB = \{\vv_1, \vv_2, \ldots, \vv_n\}\) of \(\R^n\) consisting of eigenvectors of \(A\text{.}\) Assume that \(A \vv_i = \lambda_i \vv_i\) for each \(i\text{.}\) Letting \(P = [\vv_1 \ \vv_2 \ \ldots \ \vv_n]\) we know that

where \(D\) is the diagonal matrix with \(\lambda_1\text{,}\) \(\lambda_2\text{,}\) \(\ldots\text{,}\) \(\lambda_n\) down the diagonal. If \(T\) is the matrix transformation defined by \(T(\vx) = A\vx\text{,}\) then

Now \(P^{-1}\) is a change of basis matrix from the standard basis to the basis \(\CB\text{,}\) and \(D\) stretches or contracts space in the direction of \(\vv_i\) by the factor \(\lambda_i\text{,}\) and then \(P\) changes basis back to the standard basis. In this way we can visualize the action of the matrix transformation using the basis \(\CB\text{.}\) If \(A\) is not a diagonalizable matrix, we can use the Jordan canonical form to understand the action of the transformation defined by \(A\text{.}\) We start by analyzing shears.

As we did with diagonalizable matrices, we can understand a general matrix transformation of the form \(T(\vx) = A\vx\) by using a Jordan canonical form of \(A\text{.}\) In this context, we will encounter matrices of the form \(B = \left[ \begin{array}{cc} c\amp a\\0\amp c \end{array} \right] = \left[ \begin{array}{cc} c\amp 0 \\ 0\amp c \end{array} \right] \left[ \begin{array}{cc} 1\amp \frac{a}{c} \\ 0\amp 1 \end{array} \right]\) for some positive constant \(c\text{.}\) If \(S(\vx) = B\vx\text{,}\) then \(S\) performs a shear in the direction of \(\ve_1\) and then an expansion or contraction in all directions by a factor of \(c\text{.}\) We illustrate with an example.

Activity 40.7 provides the essential ideas to understand the geometry of a general linear transformation using the Jordan canonical form. Let \(T(\vx) = A\vx\) with \(A \vv_1 = \lambda \vv_1\) and \(A \vv_2 = \lambda \vv_2 + \vv_1\text{.}\) Then \(T\) maps \(\Span \{\vv_1\}\) to \(\Span \{\vv_1\}\) by a factor of \(\lambda\text{,}\) and \(T\) maps \(\Span\{\vv_2\}\) to the line containing the terminal point of \(\vv_1\) in the direction of \(\vv_2\text{.}\) The matrix \(C\) performs a change of basis from the standard basis to the basis \(\{\vv_1, \vv_2\}\text{,}\) then the matrix \(\left[ \begin{array}{cc} 2\amp 1\\0\amp 2 \end{array} \right]\) performs an expansion by a factor of 2 in all directions and a shear in the direction of \(\vv_1\text{.}\)

Subsection Proof of the Existence of the Jordan Canonical Form

While we have constructed an algorithm to find a Jordan canonical form of a square matrix, we haven't yet addressed the question of whether every square matrix has a Jordan canonical form. We do that in this section.

Consider that any vector \([a \ b]^{\tr}\) in \(\R^2\) can be written as the sum \([a \ 0]^{\tr} + [0 \ b]^{\tr}\text{.}\) The vector \([a \ 0]^{\tr}\) is a vector in the subspace \(W_1 = \{ [x \ 0]^{\tr} | x \in \R\} = \Span \{[1 \ 0]^{\tr}\}\) and the vector \([0 \ b]^{\tr}\) is in the subspace \(W_2 = \{[0 \ y]^{\tr} | y \in \R\} = \Span\{[0 \ 1]^{\tr}\text{.}\) Also notice that \(W_1 \cap W_2 = \{\vzero\}\text{.}\) We can extend this idea to \(\R^3\text{,}\) where any vector \([a \ b \ c]^{\tr}\) can be written as \([a \ 0 \ 0]^{\tr} + [0 \ b \ 0]^{\tr} + [0 \ 0 \ c]^{\tr}\text{,}\) with \([a \ 0 \ 0]^{\tr}\) in \(W_1 = \Span\{[1 \ 0 \ 0]^{\tr}\text{,}\) \([0 \ b \ 0]^{\tr}\) in \(W_2 = \Span\{[0 \ 1 \ 0]^{\tr}\text{,}\) and \([0 \ 0 \ c]^{\tr}\) in \(W_3 = \Span\{[0 \ 0 \ 1]^{\tr}\text{.}\) In this situation we write \(\R^2 = W_1 \oplus W_2\) and \(\R^3 = W_1 \oplus W_2 \oplus W_3\) and say that \(\R^2\) is the direct sum of \(W_1\) and \(W_2\text{,}\) while \(\R^3\) is the direct sum of \(W_1\text{,}\) \(W_2\text{,}\) and \(W_3\text{.}\)

If \(V\) is a direct sum of subspaces \(V_1\text{,}\) \(V_2\text{,}\) \(\ldots\text{,}\) \(V_m\text{,}\) then we write

Some useful facts about direct sums are given in the following theorem. The proofs are left for the exercises.

Subsection Nilpotent Matrices and Invariant Subspaces

We will prove the existence of the Jordan canonical form in two steps. In the next subsection Lemma 40.14 will show that every linear transformation can be diagonalized in some form, and Lemma 40.15 will provide the specific Jordan canonical form. Before we proceed to the lemmas, there are two concepts we need to introduce — nilpotent matrices and invariant subspaces. We don't need these concepts beyond our proof, so we won't spend a lot of time on them.

Activity 40.8 shows that there are some matrices whose powers eventually become the zero matrix, and that there might be some connection to the eigenvalues of these matrices. Such matrices are given a special name.

Nilpotent matrices are the essential obstacle to the diagonalization process. If \(A\) is a nilpotent matrix, the smallest positive integer \(m\) such that \(A^m = 0\) is called the index of \(A\text{.}\)

A characterization of nilpotent matrices is given in the following theorem.

The proof is left to the exercises.

We have seen that if \(T: V \to V\) is a linear transformation from a vector space to itself, and if \(\lambda\) is an eigenvalue of \(T\) with eigenvector \(\vv\text{,}\) then \(T(\vv) = \lambda \vv\text{.}\) In other words, \(T\) maps every vector in \(W = \Span\{\vv\}\) to a vector in \(W\text{.}\) When this happens we say that \(W\) is invariant under the transformation \(T\text{.}\)

So, for example, every eigenspace of a transformation is invariant under the transformation. Other spaces that are always invariant are \(V\) and \(\{\vzero\}\text{.}\)

Subsection The Jordan Canonical Form

We are now ready to prove the existence of the Jordan canonical form.

An example might help illustrate the lemma.

Since each \(\Ker(T-\lambda_i I)^{s_i}\) in Activity 40.10 is \(T\) invariant, \(T\) maps vectors in \(\Ker(T-\lambda_i I)^{s_i}\) back into \(\Ker(T-\lambda_i I)^{s_i}\text{.}\) So \(T\) applied to each \(\Ker(T-\lambda_i I)^{s_i}\) provides a matrix \(B_i\text{.}\) Applying \(T\) to each \(\Ker(T-\lambda_i I)^{s_i}\) produces the matrix

where the \(B_i\) are square matrices corresponding to the eigenvalues of \(T\text{.}\) These blocks are determined by the restriction of \(T\) to the spaces \(\Ker(T-\lambda_i I)^{s_i}\) with respect to the found basis. To obtain the Jordan canonical form, we need to know that we can always choose these basis to create the correct block matrices. Lemma 40.15 will provide those details.

Lemma 40.14 tells us that \(T\) has a diagonal form with block matrices down the diagonal. To obtain a Jordan canonical form, we need to identify the correct bases for the summands of \(V\text{.}\) (Lemma is due to Mark Wildon from “A SHORT PROOF OF THE EXISTENCE OF JORDAN NORMAL FORM”.)

Notice the similarity of Lemma 40.15 to chains of generalized eigenvectors. An example might help illustrate Lemma 40.15.

Notice that if we let \(C = [[T^2(\vu_1)]_{\CS} \ [T(\vu_1)]_{\CS} \ [\vu_1]_{\CS} \ [T(\vu_4)]_{|CS} \ [\vu_4]_{|CS} \ [\vu_6]_{\CS}]\) using the vectors from Activity 40.11 we should find that

and this basis provides a matrix that produces a Jordan canonical form of \(A\text{.}\)

Lemma 40.15 provides the sequences of generalized eigenvectors that we need to make the block matrices in Jordan canonical form. This works as follows. Start, for example, with \(\vu_1\text{,}\) \(T(\vu_1)\text{,}\) \(\ldots\text{,}\)\(T^{a_1-1}(\vu_1)\text{.}\) Since \(T^s = 0\text{,}\) we know that \(T\) is nilpotent and so has only \(0\) as an eigenvalue. If \(T\) has a nonzero eigenvalue \(\lambda\text{,}\) we replace \(T\) with \(T - \lambda I\text{.}\)

Let \(\CB = \{T^{a_1-1}(\vu_1), T^{a_1-2}(\vu_1), \ldots, T^2(\vu_1), T(\vu_1), \vu_1\}\text{.}\) Then

This makes

which gives one Jordan block.

We return to Activity 40.10 to illustrate the use of Lemma 40.15.

Subsection Examples

What follows are worked examples that use the concepts from this section.

Subsection Summary

• Any square matrix \(A\) is similar to a Jordan canonical form

  \begin{equation*} \left[ \begin{array}{cccc} J_1\amp 0\amp \cdots\amp 0 \\ 0\amp J_2\amp \cdots\amp 0 \\ \vdots\amp \vdots\amp \ddots\amp \vdots \\ 0\amp 0\amp \cdots\amp J_t \end{array} \right]\text{,} \end{equation*}

  where each matrix \(J_i\) is a Jordan block of the form

  \begin{equation*} \left[ \begin{array}{ccccccc} \lambda\amp 1\amp 0 \amp \cdots\amp 0\amp 0\amp 0 \\ 0\amp \lambda\amp 1\amp \cdots\amp 0\amp 0\amp 0 \\ 0\amp 0\amp \lambda\amp \cdots\amp 0\amp 0\amp 0 \\ \amp \amp \amp \ddots\amp \ddots\amp \amp \\ 0\amp 0\amp 0\amp \cdots\amp \lambda\amp 1\amp 0 \\ 0\amp 0\amp 0\amp \cdots\amp 0\amp \lambda\amp 1 \\ 0\amp 0\amp 0\amp \cdots\amp 0\amp 0\amp \lambda \end{array} \right]\text{.} \end{equation*}

  with \(\lambda\) as a eigenvalue of \(A\text{.}\)

• A generalized eigenvector of an \(n \times n\) matrix \(A\) corresponding to an eigenvalue \(\lambda\) of \(A\) is a non-zero vector \(\vx\) satisfying

  \begin{equation*} (A - \lambda I_n)^m \vx = \vzero \end{equation*}

  for some positive integer \(m\text{.}\) If \(A\) is an \(n \times n\) matrix, then we can find a basis of \(\R^n\) consisting of generalized eigenvectors \(\vv_1\text{,}\) \(\vv_2\text{,}\) \(\ldots\text{,}\) \(\vv_n\) of \(A\) so that that matrix \(C = [\vv_1 \ \vv_2 \ \cdots \ \vv_n]\) has the property that \(C^{-1}AC\) is a Jordan canonical form.

• A vector space \(V\) is a direct sum of subspaces \(V_1\text{,}\) \(V_2\text{,}\) \(\ldots\text{,}\) \(V_m\) if every vector \(\vv\) in \(V\) can be written uniquely as a sum

  \begin{equation*} \vv = \vv_1+\vv_2+\vv_3+\cdots + \vv_m\text{,} \end{equation*}

  with \(\vv_i \in V_i\) for each \(i\text{.}\)

• A square matrix \(A\) is nilpotent if and only if \(0\) is the only eigenvalue of \(A\text{.}\) The Jordan form of a matrix \(A\) can always be written in the form \(D + N\text{,}\) where \(D\) is a diagonal matrix and \(N\) is a nilpotent matrix.

• A subspace \(W\) of a vector space \(V\) is invariant under a linear transformation \(T: V \to V\) if \(T(\vw) \in W\) whenever \(\vw\) is in \(W\text{.}\)

Subsection Project: Modeling an Epidemic

The COVID-19 epidemic has generated many mathematical and statistical models to try to understand the spread of the virus. In this project we examine a simple stochastic model of the spread of an epidemic proposed by Norman Bailey in 1950.⁶⁴ This is a model of a relatively mild epidemic in which no one dies from the disease. Of course, mathematicians build on simple models to form more complicated and realistic ones, but this is a good, and accessible, starting point. Bailey writes about the the difficulties in the stochastic⁶⁵ analysis of epidemics. For example, the overall epidemic “can often be broken down into smaller epidemics occurring in separate regional subdivisions” and that these regional epidemics “are not necessarily in phase and often interact with each other”. This is behavior that has been clearly evident in the COVID-19 epidemic in the US. Even within a single district, “it is obvious that a given infectious individual has not the same chance of infecting each inhabitant. He will probably be in close contact with a small number of people only, perhaps of the order of 10-50, depending on the nature of his activities.” But then the epidemic for the whole district will “be built up from epidemics taking place in several relatively small groups of associates and acquaintances.” So we can see in this analysis that an epidemic can spread from small, localized areas.

Bailey begins by considering a community of \(n\) persons who are susceptible to a disease, and supposes introducing a single infected individual into the community. Bailey makes the following assumptions: “We shall assume that the infection spreads by contact between the members of the community, and that it is not sufficiently serious for cases to be withdrawn from circulation by isolation or death; also that no case becomes clear of infection during the course of the main part of the epidemic.”

To see how the Bailey's model is constructed, we make some assumption. We split the population at any time into two groups, those infected with the disease, and those susceptible, i.e., not currently infected. We assume that once an individual is infected, that individual is always infected. A person catches the disease by interacting with an infected individual. That is, if a susceptible individual meets an infected individual, the chance that the susceptible person contracts the disease is \(\beta\) (the infection rate). For example, there many be a \(5\%\) chance that an encounter between a susceptible individual and an infected individual results in the susceptible individual contracting the disease. (Of course, there are many variables involved in such an interaction — if no one is wearing masks and the interaction involves close contact, the infection rate would be higher than if everyone practices social distancing. For the sake of simplicity, we assume one infection rate for the entire population.) With this simple model we assume assume homogeneous mixing in the population. That is, it is equally likely that any one individual will interact with any other in a given time frame. Let \(y\) be the number of susceptible individuals in the population at time \(t\text{.}\) Then the number of infected individuals is the total population minus \(y\text{,}\) or \(n+1-y\) (recall that we introduced an infected individual into the population). So the change in the number of susceptible individuals in the population at time \(s\) is \(-\beta y(n+1-y)\) (since the \(n+1-y\) susceptible individuals interact with the \(y\) infected individuals). That is, \(\frac{dy}{ds} = -\beta y (n+1-y)\text{.}\) Bailey makes the substitution of \(t\) for \(\beta s\) to simplify the equation. This substitution makes \(ds = \frac{1}{\beta} dt\text{,}\) which produces the differential equation \(\beta \frac{dy}{dt} = -\beta y (n+1-y)\text{.}\) The \(\beta\)s cancel, producing the differential equation

with \(y(0) = n\text{.}\)

The above analysis is just to provide some background. Bailey's stochastic model deals with probabilities instead of actual numbers, so we now take that approach. For \(r\) from \(0\) to \(n\text{,}\) let \(p_r(t)\) be the probability that there are \(r\) susceptible individuals still uninfected at time \(t\text{.}\) Similar to the case describe above, if there are \(r\) susceptible individuals, any one of them can be come infected by interacting with the \(n-r+1\) infected individuals. With that in mind, Bailey's model of the spread of the disease is the system

That is,

Since we start with \(n\) susceptible individuals, at time \(0\) we have \(p_r(0) = 0\) if \(0 \leq r \leq n-1\) and \(p_n(0)=1\text{.}\)

This system can be written in matrix form. Let \(P(t) = \left[ \begin{array}{c} p_0(t)\\p_1(t) \\ \vdots \\ p_n(t) \end{array} \right]\text{.}\) Assuming that we can differentiate a vector function component-wise, our system becomes

with initial condition \(P(0) = [0 \ 0 \ \ldots \ 0 \ 1]^{\tr}\text{,}\) where

The solution to this system involves a matrix exponential \(e^{At}\text{.}\) The matrix exponential acts much like our familiar exponential function in that

With this in mind it is not difficult to see that Bailey's system has solution \(P(t)= e^{At}P(0)\text{.}\) To truly understand this solution, we need to make sense of the matrix exponential \(e^{At}\text{.}\)

We can make sense of the matrix exponential by utilizing the Taylor series of the exponential function centered at the origin. From calculus we know that

Since powers of square matrices are defined, the matrix exponential \(e^M\) of a square matrix \(M\) is then

Just as in calculus, \(e^M\) converges for any square matrix \(M\text{.}\)

As we will see, the matrix \(A\) in Bailey's model is not diagonalizable, so we need to learn how to consider the matrix exponential in this new situation. In these cases we can utilize the Jordan canonical form. Of course, the computations are more complicated. We first illustrate with the \(2 \times 2\) case.

Now we turn to the general case of finding the matrix exponential \(e^M\) when \(M\) is not diagonalizable. Suppose \(C\) is an invertible matrix and \(C^{-1}MC = J\text{,}\) where \(J\) is a Jordan canonical form of \(M\text{.}\) Then we have

We know that \(J\) can be written in the form \(D+N\text{,}\) where \(D\) is a diagonal matrix and \(N\) is an upper triangular matrix with zeros on the diagonal and some ones along the superdiagonal. It follows that

We know that if

then

So finding \(e^M\) boils down to determining \(e^N\text{.}\)

Now \(N\) has only zero as an eigenvalue, so \(N\) is nilpotent. If \(k\) is the index of \(N\text{,}\) then

One use for mathematical models like Bailey's is to make predictions that can help set policies. Recall that we made the substitution of \(t\) for \(\beta s\) in our original equation in order to make the equations dimensionless, where \(\beta\) is the infection rate — the rate at which people catch the disease. Let us replace \(t\) with \(\beta s\) in our solution and analyze the effect of changing the value of \(\beta\text{.}\)

In general, the matrix \(A\) has eigenvalues of algebraic multiplicity 1 and 2. When \(n\) is even, \(n+1\) is odd and the eigenvalues of \(A\) will occur in pairs except for the single eigenvalue \(0\) of multiplicity 1. When \(n\) is odd, \(n+1\) is even and we have two eigenvalues of multiplicity 1: \(0\) and the eigenvalue \(-r(n-r+1)\) when \(r = \frac{n+1}{2}\text{,}\) at which \(-r(n-r+1) = -\frac{(n+1)^2}{4}\text{.}\) It can be shown, although we won't do it here, that every eigenvalue of algebraic multiplicity 2 has geometric multiplicity 1. This information completely determines the Jordan canonical formof \(A\text{.}\)