Linear Algebra

1. Basic concepts

2. Matrix operations

Basic Concepts

m-dimensional column vector

n-dimensional row vector

mxn-dimensional matrix

Square matrix: m = n

3

0

1

2

2

1

bb

mb

b

b

52421 aa naaa

02

63

12

21

22221

11211

AA

mnmm

n

n

aaa

aaa

aaa

Matrix Addition & Subtraction

Only possible for matrices of same dimension Add/subtract matrices element-by-element Addition example: C = A+B

Subtraction example: C = A-B

283

537

356

234

213

125

451

324

231

609

323

334

241

335

124

Scalar and Matrix Multiplication

Scalar multiplication» B = kA» Dimensions: » General formula: » Example

Matrix multiplication» C = AB» Only possible if the number of columns of A is

equal to the number of rows of B

33

96

11

323

mxnmxn RRRk BA ijijij bBkab

Matrix Multiplication cont.

General representation» Dimensions:

» Formula

Examples

Noncommutative operation:

3

2

1

4

231

220

848

012

310

212

101

321

rjmibacn

iikjijk ,,1,,1

1

mxrnxrmxn RRR CBA

BAAB

Transpose

Notation: B = AT

Dimensions: Formula: Example

Important properties

nxmmxn RR BA

ijjiij bBab

642

531

65

43

21T

TTT

TT

TTT

cc

ABAB

AA

BABA

)(

Common Matrices

Symmetric matrix: AT = A Skew-symmetric matrix: AT = -A Example of a diagonal matrix

Examples of triangular matrices

Identity matrix

43

01:Lower

300

310

235

:Upper AA

200

030

001

A

AIAAII

10

01

Systems of Linear Algebraic Equations

Scalar representation

Matrix representation: Ax = b

Homogeneous system: b = 0» One obvious solution: x = 0

62

332

21

21

2211

22222121

11212111

xx

xx

bxaxaxa

bxaxaxa

bxaxaxa

mnmnmm

nn

nn

mnmnmm

n

n

b

b

b

x

x

x

aaa

aaa

aaa

2

1

2

1

21

22221

11211

bxA

Triangular Systems

Example

Solution

Gaussian elimination» Transform original system into diagonal form» Accomplished by elementary row operations

1

3

4

232

011

002

3

2

1

x

x

x

12

3211232

133

242

213321

1221

11

xxxxxx

xxxx

xx

Systems of Linear Algebraic Equations

Scalar representation

Matrix representation: Ax = b

Homogeneous system: b = 0» One obvious solution: x = 0

62

332

21

21

2211

22222121

11212111

xx

xx

bxaxaxa

bxaxaxa

bxaxaxa

mnmnmm

nn

nn

mnmnmm

n

n

b

b

b

x

x

x

aaa

aaa

aaa

2

1

2

1

21

22221

11211

bxA