Notes on Linear Odes

Remark 1.3. Suppose the characteristic polynomial is given by χA(λ) = Π m i=1(λ−λi) , where λi are distinct. Then the polynomials p`,j(t) have degree strictly smaller than νj (Exercise: Why?) If A is real, we want to find an expression for general real solutions. These are in particular the solutions one obtain when given real initial data. Before we proceed, let us just make a simple observation that if A is an (n× n) real matrix and λ ∈ C \R is an eigenvalue, then its complex conjugate λ̄ is also an eigenvalue. (This is because if Av = λv for v ∈ C \ {0}, then Av̄ = λ̄v̄.) With this observation, we have the following corollary: Corollary 1.4. Let A be an (n × n) real matrix. Suppose μ1, . . . , μm are all of its real eigenvalues and α1 ± iβ1, . . . , αs ± iβs are all of its eigenvalues in C \ R. Then the `-th component of any real solution to u′(t) = Au(t) takes the following form