msw-lecture 2

I.1 Mathematical Review
I.1.1 Basic Concepts I.1.2 Linear Ordinary Differential Equations
I.1.1
Basic Concepts 1. Vector Space V Linear Space V if it satisfies with (or ),
then
∂ ∂UT ∂VT T (U V) = V+ U ∂X ∂X ∂X
Proof:
UT (X)V(X) = ∑ui (X)vi (X)
n
So
i =1
∂v ∂(UT V) n ∂ui = ∑( vi + ui i ) i =1 ∂X ∂X ∂X n ∂u n ∂v = ∑ i vi + ∑ui i i =1 ∂X i =1 ∂X
p12 (X) p22 (X) ... pn2 (X) ... p1m (X) ... p2m (X) ... ... ... pnm(X)
Furthermore, Matrix Function of Vector
P : Rr → Rn×m
Example
or
x1 x X= 2 ... xr
V, then
2. Inner Products of v&w
T T Suppose v = [v1 , v2 ,L, vn ] & w = [w1 , w2 ,L, wn ] , v ∈V, w ∈ W
w 1 w n 2 vT w = v,w = [v1, v2 ,...vn ] = v1w + v2w2 +... + vnwn = ∑vi wi 1 ... i=1 wn
11.
Derivative of row vector function
ϕ(X) = [φ1 (X),φ2 (X),L,φn (X)]
∂φ1 ∂x 1 ∂φ ∂ϕ(X) 1 = ∂x2 ∂X ... ∂φ 1 ∂xm ∂φ2 ∂x1 ∂φ2 ∂x2 ... ∂φ2 ∂xm
v)
v
v2
5. The distance of x & y
d(x, y)∆ x − y = < x - y, x - y > = ∑(xi − yi )2 i=1
n 1 2
x = [x1, x2 ,L, xn ]T & y = [ y1, y2 ,L, yn ]T , x ∈V, y ∈V
PRODUCT RULE A(t) and B(t) are both differentiable matrices, then
d & & (A(t)B(t)) = A(t)B(t) + A(t)B(t) dt
INNER PRODUCT RULE If U : Rm → Rn or
V : Rm → Rn
12. Differentiation of Matrix Function
SUM RULE If A(t) and B(t) are both differentiable matrices Proof: Let
a11 (t) + b11 (t) a (t) + b (t) 21 C(t) = A(t) + B(t) = 21 ... an1 (t) + bn1 (t)
p11(t) p (t) P(t) = 21 ... pn1(t)
p11(X) p (X) P(X) = 21 ... pn1(X)
p12(t) ... p1m (t) p22 (t) ... p2m (t) ... ... ... pn2 (t) pnm(t)
u1 (x1 , x2 ,..., xm ) v1 (x1 , x2 ,..., xm ) u (x , x ,..., x ) v (x , x ,..., x ) 2 1 2 m m U(X) = V(X) = 2 1 2 ... ... un (x1 , x2 ,..., xm ) vn (x1 , x2 ,..., xm )
4. Euclidean Norm of Vector
v
n 1 2
2 2 2 2 v = < v, v > = v1 + v2 +…+ vn = ∑vi i=1
where v = [v1 , v2 ,L, vn ]T v∈V . The geometrical meaning
two vectors v&w are orthogonal (or perpendicular) if < v, w >= vT w = 0
3. Outer Product of v&w
v1 v1w v1w2 1 v w v w v vwT = 2 [w , w2 ,...wn ] = 2 1 2 2 ... ... 1 ... vn 1 vnw vnw2 ... v1wn ... v2wn ... ... ... vnwn
f1(x1, x2 ,...xm ) f1(X) f (x , x ,...x ) f (X) 10. Jacobian matrix of f (X) = 2 1 2 m = 2 is ... ... fn (x1, x2 ,...xm ) fn (X)
Notations: ∇f (X) or grad f (X) or
∂f ∂X
Example:
x1 2 f (X) = f (x2 ) = x1 + sin x2 + ex3 x3
2x1 ∂f Then ∇f = = cos x2 ∂X x3 e
Subspace: Suppose
for all v1 & v2 in V for all v1,v2& v3 in V v,−v in V
v1 ∩v2 is also a subspace of V
v1 + v2 is also a subspace of V
v1and v2 are subspace of
1) Sums
v1 + (v2 + v3 ) = (v1 + v2 ) + v3 v + (−v) = (−v) + v = 0
v1 + v2 = v2 + v1
v +0 = 0+v = v v ∈V 2) Products r1 , r2 are real numbers, v ∈V (a) r1 ⋅ (r2 ⋅ v) = (r1r2 )v (b) r ⋅ (v1 + v2 ) = rv1 + rv2 r is a real number, v1 ,v2 ∈V (c) (r + r2 ) ⋅ v = r v + r2v r1 , r2are real numbers, v ∈V 1 1
φ(X)
∂φn ∂x1 ∂φn ... ∂x2 ... ... ∂φn ... ∂xm m×n ...
Example: If X = [x1 , x2 ,L, xm ]T then
∂X ∂XT = =I ∂X ∂X where I is a identity matrix.
Example:
x1 x 2 3 x1 + x2 − x4 2 f (X) = f ( ) = x3 sin x2 − x3 x4
f : R4 → R2
the Jacobian matrix is
2 1 0 − 3x4 ∂f 2x1 = ∂X 0 cos x2 −1 0
d & & (A(t) + B(t)) = (A(t) + B(t)) dt
... a1m (t) + b1m (t) ... a2m (t) + b2m (t) ... ... ... anm (t) + bnm (t)
then
d(a1m + b1m ) d(a11 + b11) ... dt dt dC(t) = ... ... ... d(a + b ) dt d(anm + bnm ) n1 n1 ... dt dt & & & & (a11 + b11) ... (a1m + b1m ) a11 ... a1m b11 ... b1m & & & & = ... ... ... = ... ... ... + ... ... ... & & & & (an1 + bn1 ) ... (anm + bnm ) an1 ... anm bn1 ... bnm & & & & = A(t) + B(t)
where
2x1 + 3x22 + ln x3 f (X) = sin x1 + cos x3 log x2
f1(X) = 2x1 +3x2 + ln x3
2
f2 (X) = sin x1 + cos x3
f3 (X) = log x2
x1 and X = x2 x3
f : Rm → Rn
or