2.2 Canonical 2x2 Matrices

e^{t A} = P e^{t B} P^{- 1}

If similar

Diagonal

Suppose

B = [\begin{matrix} λ_{1} & 0 \\ 0 & λ_{2} \end{matrix}], λ_{1} \geq λ_{2} .

Eigenvalues and Eigenvectors:
The matrix has two real eigenvalues $λ_{1}, λ_{2}$ with eigenvectors

v_{1} = [\begin{matrix} 1 \\ 0 \end{matrix}], v_{2} = [\begin{matrix} 0 \\ 1 \end{matrix}] .

Fundamental Set of Solutions:

x_{1} (t) = e^{λ_{1} t} v_{1}, x_{2} (t) = e^{λ_{2} t} v_{2} .

Each of these solutions lies in an invariant subspace of $B$ .

General Solution:

x (t) = c_{1} x_{1} (t) + c_{2} x_{2} (t) = [\begin{matrix} c_{1} e^{λ_{1} t} \\ c_{2} e^{λ_{2} t} \end{matrix}],

where $c_{1}, c_{2}$ are determined by the initial condition $x_{0}$ .

\begin{array}{cccc} Eigenvalues & t \to \infty & t \to - \infty & Type \\ λ_{1} > λ_{2} > 0 & x (t) diverges and bends to direction of v_{1} & x (t) converges to (0, 0) and bends to direction of v_{2} & Nodal Source \\ 0 > λ_{1} > λ_{2} & x (t) converges to (0, 0) and bends to direction of v_{1} & x (t) diverges in both direction and bends to direction of v_{2} & Nodal Sink \\ λ_{1} = λ_{2} > 0 & x (t) diverges without bending direction & x (t) converges to (0, 0) without bending direction & Star Source \\ 0 > λ_{1} = λ_{2} & x (t) converges to (0, 0) without bending direction & x (t) diverges without bending direction & Star Sink \\ λ_{1} > 0 > λ_{2} & x (t) converges to 0 in direction of v_{2} and diverges in direction of v_{1} & x (t) converges to 0 in direction of v_{1} and diverges in direction of v_{2} & Saddle \\ λ_{1} > 0 = λ_{2} & x (t) stays at rest in direction of v_{2} and diverges in direction of v_{1} & x (t) converges to 0 in direction of v_{1} and stays at rest in direction of v_{2} & Degenerate Source \\ 0 = λ_{1} > λ_{2} & x (t) stays at rest in direction of v_{1} and converges to 0 in direction of v_{2} & x (t) stays at rest in direction of v_{1} and diverges in direction of v_{2} & Degenerate Sink \\ 0 = λ_{1} = λ_{2} & Every point is staying at rest & Every point is staying at rest & Trivial \end{array}

Suppose

B = [\begin{matrix} a & b \\ - b & a \end{matrix}], b \neq 0.

Eigenvalues and Eigenvectors:
The matrix has two conjugate complex eigenvalues

λ = a + i b, \bar{λ} = a - i b,

with corresponding eigenvectors

v = [\begin{matrix} 1 \\ i \end{matrix}], \bar{v} = [\begin{matrix} 1 \\ - i \end{matrix}] .

Fundamental Set of Solutions:

x_{1} (t) = e^{a t} [\begin{matrix} \cos (b t) \\ - \sin (b t) \end{matrix}], x_{2} (t) = e^{a t} [\begin{matrix} \sin (b t) \\ \cos (b t) \end{matrix}] .

General Solution:

x (t) = c_{1} x_{1} (t) + c_{2} x_{2} (t) = e^{a t} [\begin{matrix} c_{1} \cos (b t) + c_{2} \sin (b t) \\ - c_{1} \sin (b t) + c_{2} \cos (b t) \end{matrix}],

where $c_{1}, c_{2}$ are determined by the initial condition $x_{0}$ .

\begin{array}{cccc} Eigenvalues & t \to \infty & t \to - \infty & Type \\ a > 0 & x (t) spins around (0, 0) and diverges to \infty & x (t) spins around (0, 0) and converges to (0, 0) & Spiral Source \\ a < 0 & x (t) spins around (0, 0) and converges to (0, 0) & x (t) spins around (0, 0) and diverges to \infty & Spiral Sink \\ a = 0 & x (t) spins around (0, 0), never converges to (0, 0) or diverges & x (t) spins around (0, 0), never converges to (0, 0) or diverges & Center \end{array}

Suppose

B = [\begin{matrix} λ & 1 \\ 0 & λ \end{matrix}], λ \neq 0.

Eigenvalues and Eigenvectors:
The matrix has a repeated real eigenvalue

λ_{1} = λ_{2} = λ,

with eigenvector and generalized eigenvector

v = [\begin{matrix} 1 \\ 0 \end{matrix}], \bar{v} = [\begin{matrix} 0 \\ 1 \end{matrix}] .

Fundamental Set of Solutions:

x_{1} (t) = e^{λ t} v, x_{2} (t) = e^{λ t} (t v + \bar{v}) .

Each of these two solutions lies in an invariant subspace of $B$ .

General Solution:

x (t) = c_{1} x_{1} (t) + c_{2} x_{2} (t) = e^{λ t} [\begin{matrix} c_{1} + c_{2} t \\ c_{2} \end{matrix}],

where $c_{1}, c_{2}$ are determined by the initial condition $x_{0}$ .

\begin{array}{cccc} Eigenvalues & t \to \infty & t \to - \infty & Type \\ λ > 0 & x (t) diverges to \infty, bending to the direction of v_{1} & x (t) converges to (0, 0), bending to the direction of - v_{1} & Defective Source \\ λ < 0 & x (t) converges to (0, 0), bending to the direction of v_{1} & x (t) diverges, bending to the direction of - v_{1} & Defective Sink \end{array}

det (A) = λ_{1} λ_{2}, Tr(A) = λ_{1} + λ_{2}