Upload
harold-bridges
View
229
Download
0
Embed Size (px)
Citation preview
MA2213 Review Lectures 1-4
Inner Products
Gramm Matrices
Gram-Schmidt Orthogonalization
Transpose and its Properties
Transpose
2
3
74
7
23483
83
T
TTT MNMN )(TT MMM )()(0det 11
and
Proofs
MM T detdet
MM T is positive definite
0)()()(0 MvMvvMMvv TTT
Theorem 1
IIMMMM TTTT )()( 11
Inner ( = Scalar) Product Spaces
is a vector space RVvuRvu ,,),(
symmetry
V over realswith an inner productthat satisfies the following 3 properties:
),(),( uvvu linearityVwvuRbawubvuabwavu ,,,,),,(),(),(
positivity 0),(0 uuu
Remark Symmetry and Linearity imply ),(),(),(),(),(),( uwbuvawubvuabwavuubwav
hence (- , -) : V x V R is Bilinear
Examples of Inner Product Spaces
Example 2.( is called a weight function) and
),0(),(:]),,([ babaCV
Example 1.nnRP positive definite, symmetric
Remark The standard inner product on
VvuPvuvuRV Tn ,,),(,
nRV is obtained by choosing IP then
j
n
j jTT vuvuIvuvu
1
),(
b
adssvsusvu )()()(),( Remark The SIP
on ]),([ baC is obtained by choosing 1
The Gramm Matrix
nvv ,...,1
matrix
mjn RvvvB ],,...,[ 1
of a finite sequence of vectors
in an inner product space V is the
Theorem 2 Let
is the Gramm matrix of the
)B
njijiijnn vvGRG ,...,1,),,(,
Then BBTnvv ,...,( 1 are the columns vectors of
sequence nvv ,...,1 Proof ijTBB )(
),()(11 ji
n
k kjki
n
k kjikT vvBBBB
Standard Basis
nee ,...,1
Definition The standard (sequence)
of basis vectors for nR is
where
0
0
0
0
1
1
e
0
0
0
1
0
2
e
0
1
0
0
0
1
ne
1
0
0
0
0
ne
Questions
nn
nn ReeRA ,...,, 1
Question 1. What is the following matrix
nnnn ReeeeeM
1321
Question 2. What is the following ?
?jTi Aee if
Question 2. For the standard inner
product on ,nR what is ),( ji ee ?
Gram-Schmidt OrthogonalizationTheorem 3. Given a sequence of linearly independent vectors in an innerproduct space
nvvv ,...,, 21there exists a unique upperV
Proof Since
triangular matrix U with diagonal entries 1Uvvvuuu nn ][][ 2121 such that the ‘matrix’
has orthogonal column vectors.
nivLvui
k kikii ...,1,1
1
it suffices to show that njiuv ji ,0),(1,...,1),,(),(
1
1
jivvLvv ji
j
k jkki
these are n-1 systems with Gramm matrices
for nj ,...,2
Gram-Schmidt Algorithm
nvvv ,...,, 21start with
11 vu
111
2122 ),(
),(u
uu
vuvu
111
312
22
3233 ),(
),(
),(
),(u
uu
vuu
uu
vuvu
111
11
11
1
),(
),(
),(
),(u
uu
vuu
uu
vuvu n
nnn
nnnn
Gram-Schmidt Orthonormalization
nvvv ,...,, 21start with
||||/, 11111 uuqvu
||||/,),( 22212122 uuqqvqvu
||||/,),(),( 33313123233 uuqqvqqvqvu
,),(),( 1111 qvqqvqvu nnnnnn
1||||/ nnn uuq
produce an orthonormal basis
Here ),(|||| vvv