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