Linear Systems and Eigenvalues

Definition A first-order linear system for $n$ unknown functions $x_1(t), \dots, x_n(t)$ is

where $\mathbf{x} = \langle x_1, \dots, x_n \rangle$ is written as a column, $A(t)$ is an $n \times n$ matrix of functions and $\mathbf{g}(t)$ is a vector of functions. It is homogeneous when $\mathbf{g} = \mathbf{0}$ and has constant coefficients when $A$ does not depend on $t$. If the entries of $A$ and $\mathbf{g}$ are continuous on an open interval $I$ containing $t_0$, then for every vector $\mathbf{x}_0$ the initial-value problem with $\mathbf{x}(t_0) = \mathbf{x}_0$ has exactly one solution, and it exists on all of $I$.

Definition For an $n \times n$ real matrix $A$ and a vector $\mathbf{x}$ in $\mathbb{R}^n$, the product $A\mathbf{x}$ is the vector whose $i$th entry is $\sum_{j} a_{ij} x_j$. The map $\mathbf{x} \mapsto A\mathbf{x}$ is linear. For the homogeneous system $\mathbf{x}' = A\mathbf{x}$ with $A$ constant, every linear combination of solutions is again a solution, and the solutions form a vector space of dimension $n$. Solutions $\mathbf{x}_1, \dots, \mathbf{x}_n$ form a fundamental set when their Wronskian

is nonzero. For solutions of the same system on an interval, $W$ is either zero everywhere or zero nowhere, and for a fundamental set every solution equals $c_1 \mathbf{x}_1 + \dots + c_n \mathbf{x}_n$ for unique constants.

Definition A number $\lambda$, real or complex, is an eigenvalue of the $n \times n$ matrix $A$ if $A\mathbf{v} = \lambda\mathbf{v}$ for some nonzero vector $\mathbf{v}$, called an eigenvector for $\lambda$. The eigenvalues are the roots of the characteristic equation $\det(A - \lambda I) = 0$. For a $2 \times 2$ matrix with trace $T$ and determinant $D$ this equation is

If $A\mathbf{v} = \lambda\mathbf{v}$, then $\mathbf{x}(t) = e^{\lambda t}\mathbf{v}$ solves $\mathbf{x}' = A\mathbf{x}$, since both sides equal $\lambda e^{\lambda t}\mathbf{v}$.

Definition The matrix $A$ is diagonalizable if it has $n$ linearly independent eigenvectors $\mathbf{v}_1, \dots, \mathbf{v}_n$, with eigenvalues $\lambda_1, \dots, \lambda_n$. This holds in particular when the $n$ eigenvalues are distinct. Then $A = P\Lambda P^{-1}$ with $P = [\,\mathbf{v}_1 \cdots \mathbf{v}_n\,]$ and $\Lambda$ diagonal, and the general solution of $\mathbf{x}' = A\mathbf{x}$ is

When $A$ is real and $\lambda = \alpha + i\beta$ with $\beta \ne 0$ has eigenvector $\mathbf{v}$, the real and imaginary parts of $e^{\lambda t}\mathbf{v}$ are two linearly independent real solutions, and they replace the pair for $\lambda$ and $\bar\lambda$.

Definition Suppose the real $2 \times 2$ matrix $A$ has a double eigenvalue $\lambda$, which is then real. There are two cases. If $A = \lambda I$, every nonzero vector is an eigenvector and every solution is $e^{\lambda t}\mathbf{x}_0$. Otherwise $A$ is defective, with only one independent eigenvector $\mathbf{v}$. In that case $(A - \lambda I)^2 = 0$, so the column space of $A - \lambda I$ is the line spanned by $\mathbf{v}$, and there is always a vector $\mathbf{w}$ with $(A - \lambda I)\mathbf{w} = \mathbf{v}$, called a generalized eigenvector. Then

form a fundamental set of solutions. For an $n \times n$ matrix, an eigenvalue of multiplicity $m$ with fewer than $m$ independent eigenvectors produces solutions of the form $e^{\lambda t}$ times a vector polynomial in $t$ of degree less than $m$.

Definition For any $n \times n$ matrix $A$ and real $t$,

a series that converges for every $A$ and $t$. It satisfies $\frac{d}{dt}e^{At} = Ae^{At}$ and $e^{A \cdot 0} = I$, so the unique solution of $\mathbf{x}' = A\mathbf{x}$ with $\mathbf{x}(0) = \mathbf{x}_0$ is $\mathbf{x}(t) = e^{At}\mathbf{x}_0$. If $A = P\Lambda P^{-1}$ with $\Lambda$ diagonal, then $e^{At} = Pe^{\Lambda t}P^{-1}$. The rule $e^{A + B} = e^A e^B$ holds when $AB = BA$ and can fail otherwise.

Definition For a $2 \times 2$ real system $\mathbf{x}' = A\mathbf{x}$, the phase plane is the $x_1x_2$ plane, and the phase portrait is the family of solution curves drawn there with arrows for increasing $t$. If $\det A \ne 0$, the origin is the only equilibrium, and the eigenvalues classify it. A node, star node or spiral is stable when the real parts are negative and unstable when they are positive.

Definition The equilibrium $\mathbf{0}$ of $\mathbf{x}' = A\mathbf{x}$ is stable if for every $\varepsilon > 0$ there is $\delta > 0$ such that $\lVert \mathbf{x}(0) \rVert < \delta$ implies $\lVert \mathbf{x}(t) \rVert < \varepsilon$ for all $t \ge 0$. It is asymptotically stable if it is stable and every solution tends to $\mathbf{0}$ as $t \to \infty$, and unstable if it is not stable. For constant real $A$,

• if every eigenvalue has negative real part, $\mathbf{0}$ is asymptotically stable,
• if some eigenvalue has positive real part, $\mathbf{0}$ is unstable,
• if every real part is at most $0$ and some is $0$, then $\mathbf{0}$ is stable but not asymptotically stable exactly when each eigenvalue with zero real part has as many independent eigenvectors as its multiplicity, and unstable otherwise.