Some useful linear algebra
Linearly independent vectors
•
1v1  2v2  3v3    i vi  0 only for 1  2    i  0
• span(V): span of vector space V is all linear
combinations of vectors vi,i.e.   v
i i
i
Ax  x  (I  A) x  0 hence (I  A) is singular
The eigenvalues of A are the roots of the
characteristic equation
p( )  det( I  A)  0
diagonal form of matrix
1




2

S 1 AS    


.


N 

Eigenvectors of A are columns of S
Similarity transform
If B  M 1 AM
then A and B have the same eigenvalues
The eigenvector x of A corresponds to the eigenvector
M-1x of B
Rank and Nullspace
A
xb



  

  

  
m  n n 1
 
 
 
 
m 1
Least Squares
Ax  b
• More equations than unknowns
• Look for solution which minimizes ||Ax-b|| = (Ax-b)T(Ax-b)
• Solve ( Ax  b)T ( Ax  b)
xi
0
• Same as the solution to
• LS solution
A Ax  A b
T
T
1
x  ( A A) A b
T
T
Properties of SVD
si2 are eigenvalues of ATA
Columns of U (u1 , u2 , u3 ) are eigenvectors of AAT
Columns of V (v1 , v2 , v3 ) are eigenvectors of ATA
|| A || F   ai2, j   s i2
i, j
Solving
A  ( At A) 1 At
A1  V  1 U

with
1
x  ( A A) A b
t
0
1
t
pseudoinverse of A
and
A  V  01 U
1

equal to
for all nonzero singular
values and zero otherwise
Least squares solution of
homogeneous equation Ax=0
Minimize || Ax || subject to || x || 1
A  UDV T
|| UDV T x |||| DV T x || and || x |||| VT x ||
y  V T x  minimize || DV T x || subject to || VT x || 1
or || Dy || subject to || y || 1
diagonal elements of D in descending order
 0
 
 0
y     x  Vy  last column of V
.
 
 1
 
Enforce orthonormality
constraints on an estimated
rotation matrix R’
R'  UDV T
replace by R  UIV T
I is identity matrix
Newton iteration
X  f (P)
measurement
parameter


Find P satisfying X  f(P)  ε ||  || Min
Start with P0
Assume f is locally linear
X
f(P1 )  f ( P0 )  J
wh ere J 
P
Pi 1  Pi   i
 i is solution t o JΔi   i  X  f ( Pi )
J T J i  J T  i
f( ) is nonlinear
Levenberg Marquardt iteration
replace NΔ  J T J  J T  i
by N  with N i,i  (1   ) N i ,i
Ni,j  N i , j for i  j
Start with   10 3
if error is reduced  new   10
if error is increased  new   10 1