the matrix exponential by Laplace transform
/ luh-PLAHSS /
The Laplace transform is the algebra machine that turns calculus problems into ones you can solve by pushing symbols around. It already conquers a single scalar equation by changing derivatives into multiplication by s. The same machine, applied to the whole system x' = A x at once, hands you the matrix exponential through a single algebraic inverse — no eigenvectors, no series, no Jordan blocks.
Here is how it falls out. Transform the matrix initial value problem x' = A x, x(0) = x0. The derivative rule turns x' into s X(s) - x0, so the equation becomes s X(s) - x0 = A X(s). Gather the X terms: (sI - A) X(s) = x0, hence X(s) = (sI - A)^(-1) x0. Comparing with x(t) = e^(At) x0 reveals the clean identity: e^(At) is the inverse Laplace transform of (sI - A)^(-1). That object (sI - A)^(-1) is called the resolvent of A; its entries are rational functions of s, and you invert them back to functions of t by the ordinary tools — partial fractions and a transform table — exactly as for a single equation. So you invert one matrix symbolically, simplify each entry into recognizable transforms, and read off e^(At) entry by entry.
This method is especially comfortable when you are already fluent in Laplace transforms, because it reuses every skill you have: the poles of the resolvent (the roots of det(sI - A) = 0) are precisely the eigenvalues, and partial fractions automatically produce the right e^(lambda t), t e^(lambda t), and sin/cos pieces, repeated roots and complex roots included, with no separate cases. The cost is the symbolic inversion of an n-by-n matrix of polynomials, which grows heavy for large n; for 2-by-2 and 3-by-3 it is clean and dependable.
For A = [0, 1; -1, 0], sI - A = [s, -1; 1, s], whose determinant is s^2 + 1. The inverse is (1/(s^2+1)) [s, 1; -1, s]. Inverting each entry: s/(s^2+1) transforms back to cos t and 1/(s^2+1) to sin t, giving e^(At) = [cos t, sin t; -sin t, cos t]. The denominator s^2 + 1 is det(sI - A), whose roots i, -i are the eigenvalues — the poles announce them.
The resolvent (sI - A)^(-1) is the s-domain face of e^(At); partial fractions on its entries hand back the time-domain matrix.
The Laplace method needs the symbolic inverse of (sI - A), an n-by-n matrix of polynomials — clean for small n but quickly unwieldy as n grows, where eigenvalue methods scale better.