Why Are We Matrices? Studying plenty Matrices have uses in - - PDF document

Topic 7:

Matrices

• CSc

• p.

Why Are We Studying Matrices?

Matrices have plenty

• f

uses in Computer Science. E.g.:

• Representation...
• .

. .

• f

the graph data structure (see CSc 345)

• ...
• f

functions and relations (see Topics 8 and 9)

• Affine

transformations in Computer Graphics

• CSc

• p.

Matrix Fundamentals (1 / 3)

Definition: Matrix

[u3

• - (0L

\i\

\[') rz)

,[1

,r[ I I \

fn0jT\x

NotationT

tr) h= lt^

i5

3xz

• CSc

• p.

A l{"\cr,t{Iil\ lA An n

• d\,Lnn

{ns\on 0J! c

• [n

[tton? va t ur.a

Matrix Fundamentals (2 I 3)

Definition: Square Matrices

t\0r\ritr,t ]n 6rV\th f tr,U fVttnribv

• f fornrS

€9vAl,

f l.\ {\uWD tr &f coL'Jrnnt

Definition: Matrix Equality

Matrr C4p A-t B N{-( {(4un\ l+ truq sl"r0\t\ th.{ J(^yh-u dtrrAA{.n lrowA clnrtr (,t]'L\,r goir 01 cCI{'Nn p

• n
• l'V}$

Q,\t trt'"tMJ 1A {(vAI

• CSc

• p.

Matrix Fundamentals (3 / 3)

Definition : Transposition

lnt 1{trnsp0fttrovr Of s,A fnxn t\\et f i x Ai )vr urhnch rh.t iowj

qfU.n'l'rgAd

rY\AJ /rX ,q n ftA Ar r'rr and colurrrJ 4K

Definition: Matrix Symmetry

I\\rMl \'K fl t^ (qtnflt0"1{i q- ie A. 4r ltnox rq fwur]

bq rq v f\rq-

uil

p=F?

PTa

rl

• p.

Matrix Operations (1 I 5)

1. Matrix Addition Definition: Matrix Addition

The

• r4rvl
• .f *Wo

n X nn

\;tl,v nX M nnftt-ili X

Cii -- &i \ tbri

l^na|ri c?5

C suct,l"

h,tt q

e

;l

I

$= fs 3-l U2)

A+R =

• CSc

• A 6

Matrix Operations (2 I 5)

2. Scalar Product Definition: Scalar

A rcaf qr is ct r@L nWnbr 6*41,r",5 Cur,te,

Definition: Scalar Product

^ A Ccticif .-l e

nx'/n r/tq[ix E

" o.fr

*lne

trc 11

we prq

n^wix A +l^ql Ib1

3l

ff=tto-

7

+ft=

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Matrix Operations (3 / 5)

3. Matrix Product

llne pfbCr,tcf

nXC'nnfr+r-r,X

= g,B,n *

cil = 7,

Definition: Matrix Product (a.k rix Multiplication)

• p.

Matrix Operations (4 I 5)

ft= ffi)

F)@S n\

AF= 9Lw3o)

• CSc

• p,

Matrix Operations (5 / 5)

• r\

r^

th t/I\l

)Y)\yI lc-^ { rr'\I C

ILFZ :

W 2 urddtnod

n

I-b

• i

l-7 -l I

Ir Q, I l_k? r)

3xZ

• CSc

• p.

The ldentity Matrix

Remember the concept

• f

Multiplicative ldentity?

l.x sK

• \{ Ais l't\[n

A

A'Ip,,-

W'"\ X n hin

t_I1_i

Irn 'A =

'{"nX rn tnXn

L_t:J_j

w\ Kn

• CSc

• p.

Definition: ldentity Matrix

lm \r|l"nf il.1 f\&f (\X \A G,n n x"n 1^'6Uri(

Pof.]\o,t{A w I tls c\owq tn\ y\ctrn d"ruoroncxl 0r^d\

t

\:$

f

• rtl. ejs

{"\Srur.\y\

t L^r rty, 0S -{"\Iruo'vurt _ \ \

Matrix Powers

Definiti an: 'rlth Matrix Power

I an t{nxm ,

rceuLl+

• f

n: I >r"c\ vd6 0CA

' I,n a*T[ { powqn (

A An)'[E the

• E;

\Ie iMq+ri/ |

ft.e n+b I^rloFl-ri1

SvcCe

G,il ,(A"4)

1-wr CWece\Lee

• I '0 t{Y1X

Yn

• CSc

• p.

: Affine Transformations (1 I 3)

Used to 'move'

• bjects

in computer graphics. , Background:

t$m\oh$n

X' = Y | -f x

\'\+r$

.X

'q

3

L

I

__A (r, : )

(x,r 9,)

stc[r*,\^-.

X't sx u( " 5"1

tl .

3

L

I

LjLts

tz/

: Affine Transformations (2 I 3)

Task:

t,t) scru\( [

Sx=SlrZ

4

tz) *rcmsr6\H d A

tx:f

{q' 3

4

• p.

: Affine Tiansformations (3 / 3)

Fxo0l

ut'?\

[f

fzo

loL

LO O

L{ lrl ftt frail fql a[T]

ttnnr\oiluw

Sc6rrnol'.

tx

.t.1

0t

il

ili

\t 'O) ly'oxOt )U

• l \

t\,5)

• p.

Zero-One Matrices (1 I 3)

Three Operations: 1. 'Join': 2. 'Meet':

r\: t\\l N.Llil

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