### Finding the Eigenstructure of A

Starting with the defining equation for an eigenvector and its corresponding eigenvalue ,We normalized the first element of to 1 since is an eigenvector whenever is. (If there is a missing solution because its first element happens to be zero, we can repeat the analysis normalizing the second element to 1 instead.) Equation (G.23) gives us two equations in two unknowns:

Substituting the first into the second to eliminate , we get

and so this is the range of corresponding to sinusoidal oscillation. For , the eigenvalues are real, corresponding to exponential growth and decay. The values yield a repeated root (dc or oscillation). Let us henceforth assume . In this range is real, and we have , . Thus, the eigenvalues can be expressed as follows:

**Next Section:**

Choice of Output Signal and Initial Conditions

**Previous Section:**

Jordan Canonical Form