matrix exponential

Question 1

power series of a 1x1 matrix:

Answer 1

e^(At)=(∞)Σ(k=0)((At)^k)/k!, which converges for all real or complex values of A and for any t

Question 2

Sm:

Answer 2

(m)Σ(k-0)((At)^k)/k!

Question 3

||Sm-Sp||:

Answer 3

for any p<m, ||Sm-Sp||=||(m)Σ(k=p+1)((At)^k)/k!||<=(m)Σ(k=p+1)(||At||^k)/k! for some norm ||.||. this is a cauchy sequence so has a limit matrix, so it converges for any A and t. the exponential e^A is the case where t=1

Question 4

derivative of a matrix exponential:

Answer 4

(d/dt)e^At=Ae^At=e^(At)A, for square matrices

Question 5

e^(A+B)t=:

Answer 5

e^(At)e^(Bt) for all t iff AB=BA, for square matrices

Question 6

let A=XJX^-1 with jcl J=diag(J1,J2,…,Jp) where Ji is an mixmi jordan block with eigenvalue λi. then e^(At)=:

Answer 6

Xe^(Jt)X^-1=Xdiag(e^(J1t), e^(J2t), …, e^(Jpt))X^-1, where e^(Jit)=
e^(λit)[1 t (t^2)/(2!) … (t^(mi-1))/(mi-1)!
(t^2)/(2!)
t
1 ]
(values are in diagonal stripes)

Question 7

linear homogeneous initial value problem:

Answer 7

ẏ(t)=Ay(t), y(0)=y0 - ẏ denotes the vector whose components are the derivatives dyi/dt
y(t)=e^(At)y0
more generally with the system ẏ(t)=Ay(t)+f(t,y), y(0)=y0, y(t)=e^(At)y0 + t∫0 e^(A(t-s))f(s,y) ds

Question 8

nth order homogeneous initial value problem

Answer 8

differential equation - x^(n)(t) +a(↓n-1)x^(n-1)(t) +…+ a1ẋ(t) +a0x(t) = 0
initial values - x(0)=b1, ẋ(0)=b2, … x^(n-1)(0)=bn (where x^i denotes the ith derivative of x)
derivatives - x1(t)=x(t), xk(t)=x^(k-1)(t), k=2,…,n
system of n first order equations - ẋk(t)=x(↓k+1)(t), k=1,…,n-1, ẋn(t)=-a(↓n-1)xn(t)-…-a1x2(t)-a0x1(t)
matrix vector form - ẏ(t)=Ay(t), A=an nxn matrix with all 0 elements except the diagonal above the main diagonal which are all 1s, and the bottom row is -a0 to -a(↓n-1), where y(t)=[x1(t)…xn(t)]^T

Question 9

matrix vector form and jcl:

Answer 9

taking a matrix from a matrix vector form, let it have p distinct eigenvalues λ1,…,λp, where λi is of algebraic multiplicity mi. then the jcl of A is J=diag(J1(λ1),…,Jp(λp)), where Ji(λi) is an mixmi jordan block associated with λi - essentially, every eigenvalue appears in at most one jordan block