22 Matrix exponent. Equal eigenvalues

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1 22 Matrix exponent. Equal eigenvalues 22. Matrix exponent Consider a first order differential equation of the form y = ay, a R, with the initial condition y) = y. Of course, we know that the solution to this IVP is given by yt) = e at y. However, let us apply the method of iterations to this equation. First note that instead of differential equation plus the initial conditions we can have one integral equation yt) = y + ayτ) dτ. Now we plug in the right hand side yτ) = y and find first iteration y t): y t) = y + ay dτ = y + ay t = + at)y. I plug it again in the right hand side and find the second iteration y 2 t) = y + ay τ) dτ = + at + a2 t 2 ) y. 2 In general we find y n t) = y + ay n τ) dτ = + at! + a2 t an t n ) y. n! You should recognize inside the parenthesis the partial sums for the Taylor series of e at, hence we recover again our familiar solution yt) = e at y. So what is the point about these iterations? Let us do the same trick with a system Instead of ) we can write the integral equation ẏ = Ay, y) = y. ) yt) = y + Ayτ) dτ, MATH266: Intro to ODE by Artem Novozhilov, Fall 23

2 where the integral of a vector is understood as componentwise integrals. I plug in the right-hand side y and find the first iteration y t) = y + tay = I + At)y. Similarly to the previous, we find y n t) = I + At + A2 t An t n n! ) y. The expression in the parenthesis is a sum of n n matrices, and hence a matrix itself. Therefore, it is natural to define a matrix, which is called matrix exponent, as the infinite sum of the form: e At = exp At := I + At + A2 t 2 Note that we can include scalar t to the matrix A An t n n! +... Definition. The matrix exponent e A of A is the series e A = exp A := I + A + A An n! ) To make sure that the definition makes sense we need to specify what we understand under the infinite series of matrices. I will skip this point here and just mention that series 2) converges absolutely for any matrix A, which allows us multiply this series by another matrix, differentiate it term by term, or integrate it if necessary. Matrix exponent has a lot of properties similar to the usual exponent. Here are those that I will need in the following:. As I already mentioned, series 2) converges absolutely, which means that there is a well defined limit of partial sums of this series. 2. d dt eat = Ae At = e At A. This property can be proved by term by term differentiation and factoring out A left as an exercise). Note here that both A and e At are n n matrices, and it is not obvious that Ae At = e At A. Such matrices for which AB = BA are called commuting. 3. If A and B commute, then e A+B = e A e B. In particular A and B commute if one of them is a scalar matrix, i.e., it has the form of the form λi. 4. e λit v = e λt v, for any λ R and v R n. The proof follows from the definition. 2

3 Before using the matrix exponent to solve problems with equal eigenvalues, I would like to state the fundamental theorem of linear first order homogeneous ODE with constant coefficients: Theorem 2. Consider problem ). Then this problem has a unique solution yt) = e At y. Moreover, for any vector v R n, yt) = e At v is a solution to the system ẏ = Ay Dealing with equal eigenvalues It is important to note that the matrix exponent is not that easy to calculate for each particular example. However, the expression e At v can be easily calculated for some special vectors v without the knowldge on the explicit form of e At. Example 3. For eigenvector v with the eigenvalue λ we have that To show this, express At = λit + At λit, then e At v = e λit+at λit v = by property 3 e At v = e λt v. = e λit e A λi)t v = by property 4 = e λt e A λi)t v = by definition = e λt I + A λi)t + A λi)2 t 2 ) +... v = e λt Iv + ta λi)v + t2 A λi) 2 ) v +... = by properties of the eigenvectors = e λt Iv ) = e λt v. We actually found exactly those solutions to system ẏ = Ay that can be written down using the distinct eigenvalues. Definition 4. A nonzero vector v is called a generalized eigenvector of matrix A associated with the eigenvalue with the algebraic multiplicity k, if A λi) k v =. Now assume that vector v is a generalized eigenvector with k = 2. Exactly as in the last example, we will find that e At v = e λit+at λit v = by property 3 = e λit e A λi)t v = by property 4 = e λt e ta λi) v = by definition = e λt I + ta λi) + t2 A λi) 2 + t3 A λi) 3 ) +... v 3! = e λt Iv + ta λi)v + t2 A λi) 2 v + t3 A λi) 3 ) v +... = by properties of the eigenvectors 3! = e λt Iv + ta λi)v ) = e λt I + ta λi) ) v. 3

4 Hence we found that for a generalized eigenvector v with k = 2, the solution to our system can be taken as yt) = e λt I + ta λi) ) v, which does not require much computations. The only remaining question is actually whether we are always able to find enough linearly independent generalized eigenvectors for a given matrix. The answer is positive. Hence we obtain an algorithm for matrices with equal eigenvalues: Assume that we have a real eigenvalue λ i of multiplicity 2 and we found only one linearly independent eigenvector v i corresponding to this eigenvalue if we are able to find two, the problem is solved). Then first particular solution is given by, as before, y i t) = v i e λ it. To find a second particular solution to account for this multiplicity we need to look for a generalized eigenvector that solves the equation A λ i I) 2 u i =. Note that we are looking for such u i that the previous holds and A λ i I)u i. We can always find a solution u i of this system, which is linearly independent of v i. In this case the second particular solution is given by y i+ t) = e λ it I + A λ i I)t ) u i. This case can be generalized to the case when multiplicity of eigenvalues in bigger than 2 see an example below) and when we have complex conjugate eigenvalues of multiplicity two and higher we will not need this case for the quizzes and exams). Example 5. Find the general solution to ẏ = y. 2 The eigenvalues are λ = 2 and λ 2 = multiplicity 2). An eigenvector for λ can be taken as v =. For λ 2 we find that v 2 =, 4

5 and we are short for one more linearly independent solution to form a basis for the solution set. Consider A λ 2 I) 2 u =, which has solutions and u = u 2 =. The first one is exactly v 2, therefore we keep only u 2. Finally, one finds that y 3 t) = e t I + A λ 2 I)t ) t u 2 = e t and the general solution is yt) = C v e 2t + C 2 v 2 e t + C 3 y 3 t). Example 6. Solve the IVP 2 3 y = 2 y, y) =. 4 I find that λ = 2 is the only eigenvalue of multiplicity 3. Its eigenvector is v =, and a first linearly independent solution is given by y t) = e 2t. To find two more linearly independent solutions we need to look for the generalized eigenvectors. Consider first A λ 2 I) 2 u =, which has two vectors as a solution basis u = 5

6 and u 2 =. Note that v, u, u 2 are linearly dependent why?) and therefore we can keep only one of the vectors, e.g., u. The particular solution in this case y 2 t) = e 2t I + A 2I)t ) t u = t e 2t. t To find one more linearly independent solution let us look for a generalized eigenvector with k = 3: A λ 2 I) 3 w =. Note that A λ 2 I) 3 = therefore any vector w will do, the only thing is that we need it to be linearly independent of v and u. For instance we can take w =. Then the last solution is given by y 3 t) = e 2t I + A 2I)t + 2 A 2I)2 t 2 ) w = 3t + t2 2 2t t2 2 e 2t, + 2t t2 2 and hence the general solution to our problem is y t) t 3t + t2 y = y 2 t) = C e 2t + C 2 t e 2t 2 + C 3 2t t2 2 e 2t. y 3 t) t + 2t t2 2 Applying the initial conditions, one finds C = C 3 =, C 2 = and finally the solution is t yt) = t t e 2t How to find e At Recall that if we are given system ẏ = Ay, 3) then its solution space is an n-dimensional vector space and has a basis y,..., y n. The matrix Φt), which has y i t) as its i-th column is called the fundamental matrix solution: Φt) = y t)... y n t) ). 6

7 Theorem 7. Let Φt) be a fundamental matrix solution of 3), then e At = Φt)Φ ). Proof. First note that if Φt) is a fundamental matrix solution it solves the matrix differential equation Φ = AΦ. Moreover, since the columns of Φt) are linearly independent, then det Φ). Now, since d dt eat = Ae At, e A = I, then e At is a fundamental matrix solution itself. For any two fundamental matrix solutions Xt) and Y t) it is true that Xt) = Y t)c, where C is a constant matrix. The last equality is true since each column of Xt) can be expressed as a linear combination of columns of Y t). Therefore, by plugging t =, we find e At = Φt)C = C = Φ ). This approach is not usually the best to find e At and requires quite a few calculations. Example 8. Consider system 3) with A = To find the fundamental matrix solution, we find eigenvalues and eigenvectors of A: e t e 3t e 5t Φt) = 2e 3t 2e 5t. 2e 5t Next, Finally, /2 Φ ) = /2 /2. /2 e t e At = Φt)Φ 2 et + 2 e3t 2 e3t + 2 e5t ) = e 3t e 3t + e 5t. e 5t 7

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