5.10 Computation of the State Transition Matrix e ^At

Let A be an n n matrix, and I be the n n identity matrix. By definition, the eigenvalues ? _i, i = 1,2,..., n of A are the roots of the nth order polynomial

We recall that expansion of a determinant produces a polynomial. The roots of the polynomial of (5.133) can be real (unequal or equal), or complex numbers.

Evaluation of the state transition matrix e ^At is based on the Cayley-Hamilton theorem. This theorem states that a matrix can be expressed as an ( n ? 1) th degree polynomial in terms of the matrix A as

where the coefficients a _i are functions of the eigenvalues ?. We accept (5.134) without proving it. The proof can be found in Linear Algebra and Matrix Theory textbooks.

Since the coefficients a _i are functions of the eigenvalues ?, we must consider the following cases:

Case I: Distinct Eigenvalues (Real or Complex)

• If ? ₁ ? ? ₂ ? ? ₃ ?...? ? _n, that is, if all eigenvalues of a given matrix A are distinct, the coefficients a _i are found from the simultaneous solution of the following system of equations: