Section 3.9 The Matrix Exponential
Consider the linear system
The matrix associated with this system is
The characteristic polynomial of is
hence, there is only a single eigenvalue Moreover, we can only find a single linearly independent eigenvector Thus,
is a solution to our system. However, this is not the general solution to the system. We can only solve initial value problems where the initial condition lies on the line through the origin containing the vector To construct a general solution to our system, we will need two other linearly independent solutions. One way of doing this is with the matrix exponential.
Subsection 3.9.1 The Exponential of a Matrix
Our goal is to construct a solution to the initial value problem
where is an matrix. Recalling that the solution to the initial value problem
is we might guess that a solution to the initial value problem
has the form
if we can make sense of the expression
We will define the exponential of using the power series for Thus,
where is an matrix, where by convention. Each term makes sense in our definition since each is an expression of matrices; however, there are some issues surrounding the convergence of the power series. A series, even a series whose individual terms are matrix expressions, converges if and only if its partial sums,
converge. Although we shall not provide a proof, the matrix exponential converges for all
Theorem 3.9.2.
Proof.
We simply need to differentiate
term by term. However,
1
Since we are differentiating an infinite series, we still need to show that differentiating term by term is something that can be done. We will, however, leave the details to a course in advanced calculus.
Corollary 3.9.3.
Proof.
Thus, solving linear systems is simply a matter of computing matrix exponentials. The problem is that matrix exponentials may not be so easy to compute.
Example 3.9.4.
The matrix
has repeated eigenvalues If we try to compute then
It is not at all clear that this series converges to a matrix whose entries can be expressed in terms of elementary functions.
Now let us see how we can use the matrix exponential to solve a linear system as well as invent a more direct way to compute the matrix exponential.
Theorem 3.9.5.
Proof.
Since is an eigenvalue for we know that Using mathematical induction, we can show that has eigenvalue with associated eigenvector Indeed,
Hence,
The matrix exponential shares several properties with the exponential function that we studied in calculus.
Theorem 3.9.6.
Proof.
To prove (1), we can simply expand both sides of the equality in a power series,
Proving (2) is a only bit more complicated if we notice that the binomial expansion holds for matrices,
where
providing
Simply expand each series out to see that this is true. Part (3) follows directly from Part (2), since and commute.
Example 3.9.7.
Now let us compute once again for
First notice that
Since the identity matrix commutes with every matrix, we know that
We also know that by Example 3.9.1. Thus,
The matrix has repeated eigenvalue Consequently,
and is the zero matrix. Thus,
Our example suggests at the following proposition. We leave the proof of this proposition as an exercise.
Proposition 3.9.8.
Example 3.9.9.
We are now ready to return to our original system where
This matrix has a single eigenvalue It is easy to show that the only nonzero powers of are
Therefore,
Now, to compute three linearly independent solutions for we simply compute for three linearly independent vectors. We will use the standard basis vectors
Thus, the general solution to our system is
Subsection 3.9.2 Generalized Eigenvalues
Example 3.9.10.
Consider the system where
The characteristic polynomial of is
The eigenvalue has eigenvector and the eigenvalue has eigenvector Thus, we can find two linearly independent solutions in this case
Since has multiplicity two and we can find only one linearly independent eigenvector, it is not possible to apply Proposition 3.9.8 in this case.
If we consider the exponential
where and are linearly independent, our goal is to choose for which the series truncates. That is, we must look for vectors such that If then which means that is an eigenvector. Thus, must be a multiple of in this case. Since we already know that the eigenspace associated with this eigenvector has dimension one and is generated by we must consider higher powers.
Since
we have
The nullspace of this matrix has dimension two. Certainly, is in the nullspace of since it is the nullspace of We wish to find a vector that is not a multiple of the vector that is also in the nullspace of The vector will do. Now our series truncates,
We now have a general solution for our system,
If is an eigenvalue of and for some then is called a generalized eigenvector of When eigenvalues have algebraic multiplicity greater than one, we can compute extra solutions by looking for vectors in the nullspace of for The following theorem tells us that this is always possible. We leave the proof of the theorem as an exercise in linear algebra.
Theorem 3.9.11.
Suppose that is an eigenvalue of with multiplicity Then there exists an integer such that the dimension of the nullspace of is
We now have a procedure for finding linearly independent solutions corresponding to an eigenvalue of multiplicity
This procedure works for complex eigenvalues as well as real. If has eigenvector then set and
Activity 3.9.1. Solving Systems Using the Matrix Exponential.
(a)
(b)
(c)
Compute
(d)
(e)
Find the general solution to
Subsection 3.9.3 Important Lessons
-
Suppose that
is an eigenvalue of with multiplicity Then there exists an integer such that the dimension of the nullspace of is -
The procedure for finding
linearly independent solutions corresponding to an eigenvalue of multiplicity is the following.The procedure works for complex eigenvalues as well as real. If has eigenvector then set and
Reading Questions 3.9.4 Reading Questions
1.
What is the exponential of a matrix
2.
Explain what a generalized eigenvector for a matrix is.
Exercises 3.9.5 Exercises
Finding Solutions Using the Matrix Exponential.
Solving Initial-value Problems.
Solve each of initial-value problems in Exercise Group 3.9.5.9–16
17.
18.
Prove Proposition 3.9.8.
19.
Suppose that we are given the system
where is a matrix of constants. For example, the system
can be written in the form where
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