2.4 Multiple eigenvalues

Consider the IVP

$$x' = Ax, \quad x(0) = x_{0}$$ (2.5)

for the case when A has multiple eigenvalues.

Definition 2.4.1     Let $\lambda$ be an eigenvalue of the $n \times n$ matrix A of multiplicity $m \leq n$. Then for $k=1, \ldots , m$, any non-zero solution v of

$$(A-\lambda I)^{k} v=0$$

is called a generalized eigenvector of A.

Exercise 2.4.1:     Let v₁ be a non-zero solution of

$$(A-\lambda I) v = 0$$

and v₂ be a non-zero solution of

$$(A-\lambda I) v = v_{1} \, .$$

Then show that v₁ and v₂ are linearly independent generalized eigenvectors.

Lemma 2.4.1:     Let A be a real $n \times n$ matrix with real eigenvalues $\lambda_{1}, \ldots , \lambda_{n}$ repeated according to their multiplicity. Then there exist n generalized eigenvectors $\{ v_{1}, \ldots , v_{n} \}$ of A, such that $P = [ v_{1}, \ldots , v_{n} ]$ is nonsingular and A = S + N with

$$P^{-1} SP = {\rm diag} \; [\lambda_{j}, j = 1,2, \ldots , n ]... ...{and} \quad N^{k+1} = 0 \quad \mbox{for some} \quad k < n \, .$$

Proof: (omitted)

Remark:      J = P^-1 AP is called the Jordan canonical form of A.

Theorem 2.4.1:     Let A be a real $n \times n$ matrix with real eigenvalues $\lambda_{1}, \ldots , \lambda_{n}$ repeated according to their multiplicity. Then the IVP (2.5) has a unique solution given by

$$x(t) = P {\rm diag} \; [e^{\lambda_{j}t}] P^{-1} \left[ I + Nt +\cdots + \frac{N^{k}t^{k}}{k!} \right] x_{0} \, .$$

Corollary 2.4.1:     If $\lambda$ is a real eigenvalue of multiplicity n of an $n \times n$ matrix A, then

$$S = {\rm diag} \; [ \lambda ] = \lambda I, \quad N = A - \lambda I$$

and

$$x(t) = e^{\lambda t} \left[ I+Nt+ \cdots + \frac{N^{k} t^{k}}{k!} \right] x_{0}$$

is a solution of the IVP (2.5).

Example 2.4.1:     Solve the IVP (2.5) with

$$A = \left[ \begin{array}{lrrr} 0 & -2 & -1 & -1 \\ 1 & 2 & 1 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \, .$$

Solution:     It is easy to solve $\det (A-\lambda I) = 0$ and find that A has an eigenvalue $\lambda =1$ of multiplicity 4. Thus S=I

$$N=A-S= \left[ \begin{array}{rrrr} -1 & -2 & -1 & -1 \\ 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right]$$

$$N^{2} = \left[ \begin{array}{rrrr} -1 & -1 & -1 & -1 \\ 0 & ... ... & 0 \end{array} \right] \quad \mbox{and} \quad N^{3} = 0 \, .$$

Thus by Corollary 2.4.1,

$$\begin{eqnarray*}x(t) & = & e^{t} \left[ I+Nt+ \frac{N^{2}t^{2}}{2!} \right] x_{... ...\frac{t^{2}}{2} \\ 0 & 0 & 0 & 1 \end{array} \right] x_{0} \, . \end{eqnarray*}$$

In the general case, we must first determine a basis of generalized eigenvectors for Rⁿ and compute $S = P {\rm diag} \; [ \lambda_{j}] P^{-1}$ and N = A - S.

Example 2.4.2:     Solve the IVP (2.5) with

$$A = \left[ \begin{array}{rll} 1 & 0 & 0 \\ -1 & 2 & 0 \\ 1 & 1 & 2 \end{array} \right] \, .$$

Solution:     A has eigenvalues $\lambda_{1} = 1$, $\lambda_{2} = \lambda_{3} = 2$. It is easy to find the corresponding eigenvectors

$$v_{1} = \left[ \begin{array}{r} 1 \\ 1 \\ -2 \end{array} \rig... ..._{2} = \left[ \begin{array}{c} 0 \\ 0 \\ 1 \end{array} \right]$$

and one generalized eigenvector corresponding to $\lambda = 2$ $v_{3} = \left[ \begin{array}{c} 0 \\ 1 \\ 0 \end{array} \right]$ which is a solution of

$$\left[ \begin{array}{rll} -1 & 0 & 0 \\ -1 & 0 & 0 \\ 1 & 1... ...= \left[ \begin{array}{c} 0 \\ 0 \\ 1 \end{array} \right] \, .$$

Thus

$$P = \left[ \begin{array}{rll} 1 & 0 & 0 \\ 1 & 0 & 1 \\ -2 ... ... & 0 & 0 \\ 2 & 0 & 1 \\ -1 & 1 & 0 \end{array} \right] \, .$$

$$S = P \left[ \begin{array}{lll} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0... ... & 0 & 0 \\ -1 & 2 & 0 \\ 2 & 0 & 2 \end{array} \right] \, .$$

$$N = A -S = \left[ \begin{array}{rll} 0 & 0 & 0 \\ 0 & 0 & 0 \\ -1 & 1 & 0 \end{array} \right] \, , \quad N^{2} = 0 \, .$$

Thus by Theorem 2.4.1

$$\begin{eqnarray*}x(t) & = & P \left[ \begin{array}{lll} e^{t} & 0 & 0 \\ 0 & e^... ...} + (2-t)e^{2t} & te^{2t} & e^{2t} \end{array}\right] x_{0} \, . \end{eqnarray*}$$

In case of multiple complex eigenvalues, we have the following Lemma whose proof can be found in Hirsch and Smale, Appendix III.

Lemma 2.4.2     Let A be $2m \times 2m$ matrix with complex eigenvalues $\lambda_{j} = \alpha_{j} + i \beta_{j}$ and $\ol{\lambda}_{j} = \alpha_{j} - i\beta_{j}$, $j = 1, 2, \ldots , m$, repeated according to their multiplicity. Then there exist 2m generalized eigenvectors w_j = u_j + iv_j and $\ol{w}_{j} = u_{j} - iv_{j}$, $j=1, \ldots , m$ such that $P = [ u_{1} v_{1} \ldots u_{m} v_{m}]$ is nonsingular and

A = S+N

with

$$P^{-1} SP = {\rm diag} \; \left\{ \left[ \begin{array}{rl} \a... ..., , \quad N^{k} = 0 \quad \mbox{for some} \quad k \leq 2m \, ,$$

and SN=NS.

Remark:      J = P^-1 AP is called the Jordan canonical form of A.

Theorem 2.4.2:     Under the assumptions of Lemma 2.4.2, the IVP (2.5) has a solution

$$x(t) = P {\rm diag} \; \left\{ e^{\alpha_{j}t} \left[ \begin{... ...t + \cdots + \frac{N^{k-1} t^{k-1}}{(k-1)!} \right] x_{0} \, .$$

Example 2.4.3:     Solve the IVP (2.5) with

$$A = \left[ \begin{array}{lrlr} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 2 & 0 & 1 & 0 \end{array} \right] \, .$$

Solution:     A has eigenvalues $\lambda = i$, $\ol{\lambda} = - i$ of multiplicity 2.

Solve (A-iI)w=0 and get the eigenvector

$$w_{1} = \left( \begin{array}{c} 0 \\ 0 \\ i \\ 1 \end{array} ... ...ft( \begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \end{array} \right) \, .$$

Solve (A-iI) w = w₁ to get the generalized eigenvector

$$w_{2} = \left( \begin{array}{c} i \\ 1 \\ 0 \\ 1 \end{array} ... ...i \left( \begin{array}{c} 1 \\ 0 \\ 0 \\ 0 \end{array} \right)$$

$$P = \left[ \begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & ... ... 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array} \right]$$

$$S = P \left[ \begin{array}{rlrl} 0 & 1 & 0 & 0 \\ -1 & 0 & 0... ...0 & 0 \\ 0 & 1 & 0 & -1 \\ 1 & 0 & 1 & 0 \end{array} \right]$$

$$N=A-S= \left[ \begin{array}{lrll} 0 & 0 & 0 & 0 \\ 0 & 0 & 0... ... 0 \\ 1 & 0 & 0 & 0 \end{array} \right] \, , \quad N^{2} = 0.$$

Thus the solution to the IVP (2.5) is

$$\begin{eqnarray*}x(t) & = & P \left[ \begin{array}{rrrr} \cos t & \sin t & 0 & 0... ...t \cos t & -t \sin t & \sin t & \cos t \end{array}\right] x_{0}. \end{eqnarray*}$$

In the general case, we have the following result. (Hirsch/Smale, P. 133)

Lemma 2.4.3     Let A be a real matrix with real eigenvalues $\lambda_{j}$, $j = 1, \ldots , k$ and complex eigenvalues $\lambda_{k+s} = \alpha_{k+s} + i \beta_{k+s}$, $\ol{\lambda}_{k+s} = \alpha_{k+s} - i \beta_{k+s}$, $s=1, \ldots , m$, repeated according to their multiplicity, where k+2m=n. Then there exist generalized eigenvectors $v_{1}, \ldots , v_{k}$, w_k+1 = u_k+1 + i v_k+1, $\ol{w}_{k+1} = u_{k+1} - i v_{k+1}, \ldots , w_{k+m} = u_{k+m} + i v_{k+m}$, $\ol{w}_{k+m} = u_{k+m} - i v_{k+m}$, such that the matrix

$$P = [ v_{1}, \ldots ,v_{k}, u_{k+1}, v_{k+1}, \ldots , u_{k+m}, v_{k+m} ]$$

is nonsingular and

A = S+N

with

$$P^{-1} SP = {\rm diag} \; [ \lambda_{1}, \ldots , \lambda_{k}, \, B_{k+1}, \ldots , B_{k+m} ]$$

where $B_{j} = \left[ \begin{array}{rl} \alpha_{j} & \beta_{j} \\ - \beta_{j} & \alpha_{j} \end{array} \right]$, $j = k+1, \ldots , k+m$ and N^s = 0 for some $s \leq n$ and SN=NS.

Remark:     The matrix J = P^-1 AP is called the Jordan canonical form of A.

Theorem 2.4.3:     Under the assumption of Lemma 2.4.3 the IVP (2.5) has a solution

$$x(t) = P {\rm diag} \; \Biggl\{ e^{\lambda_{1}t} , \ldots , e... ...beta_{k+1}t & \cos \beta_{k+1}t \end{array}\right] , \ldots ,$$

$$e^{\alpha_{k+m} t} \left[ \begin{array}{rl} \cos \beta_{k+m} ... ...t + \cdots + \frac{N^{s-1} t^{s-1}}{(s-1)!} \right] x_{0} \, .$$

Based upon the above theorem, we arrive at the following important conclusion.

Theorem 2.4.4:     Each coordinate in the solution x(t) of the IVP (2.5) is a linear combination of functions of the form

$$t^{k} e^{\alpha t} \cos \beta t \quad \mbox{or} \quad t^{k} e^{\alpha t} \sin \beta t, \quad 0 \leq k \leq n -1 ,$$

where $\lambda = \alpha + i \beta$ is an eigenvalue of the matrix A, $\alpha \in R$, $\beta \geq 0$.