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Let A be a square matrix nXn then exp(At) can be written as

[tex]

exp(At)=\alpha_{n-1}A^{n-1}t^{n-1} + \alpha_{n-2}A^{n-2}t^{n-2} + ... + \alpha_1At + \alpha_0 I

[/tex]

where [tex]\alpha_0 , \alpha_1 , ... , \alpha_{n-1}[/tex] are functions of t.

Let define

[tex] r(\lambda)=\alpha_{n-1}\lambda^{n-1} + \alpha_{n-2}\lambda^{n-2} + ... + \alpha_1\lambda + \alpha_0 [/tex].

If [tex]\lambda_i[/tex] is an eigenvalue of At with multiplicity k, then

[tex] e^{\lambda_i }= r(\lambda_i) [/tex]

[tex] e^{\lambda_i} = \frac{dr}{d\lambda}|_{\lambda=\lambda_i} [/tex]

etc

Does anyone know any reference where it gives a proof for this theorem? I only know how to prove this theorem intuitively using the Cayley-Hamilton theorem. I need a formal proof. The book (Schaum Outline Series) that I got it only state the theorem.

This theorem will allowed me later to solve system of linear differential equations.

Please help.