(DM42) Matrix exponential

[Albert Chan]

(08-23-2023 07:55 PM)Albert Chan Wrote:  sinh(X) = (1 + X^2/3! + X^4/5! + ...) * X = sinh(g)/g * X

We can use same idea to get expm1(k*Jn), where Jn = matrix of all ones, dimensions n×n

Identity used: (Jn/n)^(integer_powers) = (Jn/n)

sinh(k*Jn) = (k*Jn) + (k*Jn)^3/3! + (k*Jn)^5/5! + ... = (1 + (k*n)^2/3! + (k*n)^4/5! + ...) * (k*Jn)

sinh(k*Jn) = sinh(k*n) * (Jn/n) = 2*sinh(k*n/2)*cosh(k*n/2) * (Jn/n)
cosh(k*Jn) − 1 = 2*sinh(k/2*Jn)^2 = 2*sinh(k*n/2)*sinh(k*n/2) * (Jn/n)

Add the 2 lines:

exp(x*Jn) - 1 = 2*sinh(n/2*x)*exp(n/2*x) * (Jn/n)

exmp1(k*Jn) = expm1(k*n) * (Jn/n)

---

Any function f with f(0) = 0, will have same pattern. (*)
Taylor series (Jn/n)^(integer_powers) = (Jn/n), which can be factored out.

f(k*Jn) = f(k*n * (Jn/n)) = f(k*n) * (Jn/n)

For examples, these will work: sin, versin, tan, tanh, log1p, ... (and its inverse function)

(*) Exception, if f cannot handle matrix argument.
Example, because Jn is not invertible, this will not work: f(x) := sin(x^2)/x