Some useful linear algebra

Linearly independent vectors

•

• span(V): span of vector space V is all linear combinations of vectors vi,i.e.

0for only0 21332211 iiivvvv

ii

iv

singularis)(hence0)( AIxAIxAx

The eigenvalues of A are the roots of the characteristic equation

0)det()( AIp

N

ASS

.2

1

1

Eigenvectors of A are columns of S

diagonal form of matrix

AMMB 1 If

Similarity transform

then A and B have the same eigenvalues

The eigenvector x of A corresponds to the eigenvectorM-1x of B

Rank and Nullspace

11

mnnm

bxA

Least Squares

bAx

• More equations than unknowns• Look for solution which minimizes ||Ax-b|| = (Ax-b)T(Ax-b)• Solve

• Same as the solution to

• LS solution

0)()(

i

T

x

bAxbAx

bAAxA TT

bAAAx TT 1)(

Properties of SVD

Columns of U (u1 , u2 , u3 ) are eigenvectors of AAT

Columns of V (v1 , v2 , v3 ) are eigenvectors of ATA

2 are eigenvalues of ATA

2

,

2,|||| i

jijiF aA

tt AAAA 1)(

UVAandUVA 10

11

with 10

equal to for all nonzero singular values and zero otherwise

1

pseudoinverse of A

Solving bAAAx tt 1)(

Least squares solution of homogeneous equation Ax=0

1||||tosubject||||Minimize xAx

||V||||||and|||||||| T xxxDVxUDV TT

TUDVA

1||V||subject to ||DV||minimize TT xxxVy T

order descendingin of elements diagonal D

VVyx ofcolumn last

1

.

0

0

y

1||y||subject to ||Dy||or

Enforce orthonormality constraints on an estimated

rotation matrix R’

matrixidentity is by replace

'

IUIVR

UDVRT

T

Newton iteration )(PfX

measurement

parameter

f( ) is nonlinear

Min|||| satisfying Find ε)Pf(XP

0 Start with P

linear locally is f Assume

P

XJJPf)f(P

ere wh)( 01

iii PP 1

iT

iT

iiii

JJJ

PfXJΔ

)( osolution t is

Levenberg Marquardt iteration

jiN

NNN

JJJNΔ

jii,j

iiii

iTT

for N

)1( withby

replace

,

,,

10 increased iserror if

10 reduced iserror if

10 Start with

1new

new

3