Matrix Proof

I want show that e^{(At)} = Se^{(\Lambda t)} S^{-1} , where S = (v_{1} v_{2}) ,  A is a 2 \times 2 matrix, and \Lambda = \begin{pmatrix} \lambda_{1} & 0 \\0 & \lambda_{2} \end{pmatrix}

I start by writing the middle sum in summation notation which gives me Se^{(\Lambda t)}S^{-1} = S( \displaystyle\sum_{k = 0}^{\infty} \frac{1}{k!}(\Lambda t)^{k})S^{-1} . Now I can use the identity S^{-1}AS = \Lambda which will then give me S(\displaystyle\sum_{k = 0}^{\infty} \frac{1}{k!}(S^{-1}AS)^{k}t^{k})(S^{-1}) . After pulling terms out of the sum, I will get SS^{-1} (\displaystyle \sum_{k = 0}^{\infty} \frac{t^{k}A^{k}}{k!})SS^{-1} . The SS^{-1} terms create an identity matrix and the middle sum is equivalent to e^{At} as shown below.

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s