(DM42) Matrix exponential

[Albert Chan]

Trivia, for A = 2x2 matrix, e^A only need its eigenvalues. (i.e. P, P^-1 not needed)

A = P D P^-1

D diagonal has eigenvalues of A = c ± g --> D-c has diagonal of ± g

Let X = A-c --> X^2 = g^2      // × identity matrix

sinh(X) = (1 + X^2/3! + X^4/5! + ...) * X = sinh(g)/g * X
cosh(X) = (1 + X^2/2! + X^4/4! + ...)      = cosh(g)      // × identity matrix

e^X = sinh(X) + cosh(X) = sinh(g)/g * X + cosh(g)

XCas> eA := ((sinh(g)/g) * (A-c) + cosh(g)) * e^c

XCas> A := [[a11,a12],[a21,a22]];
XCas> x1, x2 := eigenvalues(A) :;
XCas> c, g := [x1+x2, x1-x2]/2 :;
XCas> simplify(hyp2exp(exp(A) - eA))       // confirmed symbolically

\(\left(\begin{array}{cc}0&0\\0&0\end{array}\right)\)