An m n matrix is an rectangular array of elements with m rows and n columns:

mnm

n

aa

aa

A

1

111

Matrices

ija

denotes the element in the ith row and jth column

22

4/122

Xe

i

34010

301

242

413

X

t

t

et

te

)3sin(

)3cos(2

FE

DCA

Partitioning in Partitioning in submatricessubmatrices

20101

41302

51010

16301

32242

40413

Matrices y vectores son Matrices y vectores son fundamentales en el fundamentales en el

estudio formal de todas las estudio formal de todas las ramas de la ingenieríaramas de la ingeniería

InstrumentaciónInstrumentaciónDiseDiseño de circuitosño de circuitosComunicacionesComunicacionesMicroelectrónicaMicroelectrónica

A column vector is a matrix with n rows and 1 column

Vectors

na

a

a 1

A row vector is a matrix with 1 row and n columns

naaa ... 1

Square:

3

200

010

342

nm

A

mxn

Classification of matrices

m=n

Symmetric:

203

014

342

A

aji = aij

Upper Triangular:

200

010

342

Aaij = 0 when j < i

333231

232221

131211

aaa

aaa

aaa

A

Lower Triangular:

203

004

002

Aaij = 0 when j >i

333231

232221

131211

aaa

aaa

aaa

A

Diagonal:

200

010

002

Aaij = 0 when j i

Identity:

100

010

001

Aaii = 1

aij = 0 when j i

Sum of matrices of the same dimension:

mnmnmm

mm

baba

baba

BA

11

111111

Scalar multiplicationScalar multiplication BB = = kkAA

Dimensions: Dimensions:

ExampleExample

33

96

11

323

Matrix multiplicationMatrix multiplication

CC = = ABAB

Only possible if the number of Only possible if the number of columns of columns of AA is equal to the is equal to the number of rows of number of rows of BB

312321221121

3

11221

bababa

baci

ii

rkmjbacn

iikjijk ,,1,,1

1

BAC

3

2

1

4

231

220

848

012

310

212

101

321

examples:

Matrix multiplicationMatrix multiplicationis a non-commutative operation is a non-commutative operation

(generally) ::BAAB

Identity:

100

010

001

Aaii = 1

aij = 0 when j i

AIAAII

10

01

Vector products: Vector products: (u,v are column vectors)(u,v are column vectors)

Dot product or Dot product or inner productinner product

Outer product:Outer product:

22112

121 vuvu

v

vuuvuT

2212

211121

2

1

vuvu

vuvuvv

u

uvu T

2

22112

121 uuuuu

u

uuuuuT

Scalar product (of vectors)

The product of a row vector a and a column vector b is a scalar

a b = a1b1 + ... + anbn

n

iiibaba

1

cosbaba

vectorsorthogonalba 0

aaa

TraceTrace

The trace of a nxn matrix A is given by:The trace of a nxn matrix A is given by:

nn

n

iii aaaaATrace

...)( 22111

Properties of Matrix Properties of Matrix OperationsOperations

a)a) A+B = B+AA+B = B+A

b)b) A+(B+C) = A+(B+C) = (A+B)+C(A+B)+C

c)c) A(BC) = (AB)CA(BC) = (AB)C

d)d) A(B+C) = A(B+C) = AB+ACAB+AC

e)e) (B+C)A = (B+C)A = BA+CABA+CA

f)f) a(B+C) = a(B+C) = aB+aCaB+aC

Commutative law for Commutative law for additionaddition

Associative law for Associative law for additionaddition

Associative for Associative for multiplicationmultiplication

Left distributive lawLeft distributive law

Right distributive lawRight distributive lawDistributive law for Distributive law for

scalar multiplicationscalar multiplication

j)j) (a+b)C = aC+bC(a+b)C = aC+bCk)k) a(bC) = (ab)Ca(bC) = (ab)Cl)l) a(BC) = (aB)Ca(BC) = (aB)C

TransposeTranspose

BB = = AATT

Dimensions:Dimensions:

Formula:Formula:

ExampleExample

ijjiij bBab

642

531

65

43

21T

Alternative notation used in Alternative notation used in some bookssome books

BB = = AATT

BB = = AA’’

In this course we use the first one (B = AT )

Transpose Matrix Transpose Matrix propertiesproperties

TTT

TT

TTT

TT

cc

ABAB

AABABA

AA

)(

)(

Symmetric matrix: Symmetric matrix: AATT = = AA

Skew-symmetric matrix: Skew-symmetric matrix: AATT = - = -AA

Unitary matrix example :Unitary matrix example :

21

21

21

21

A

SymmetricSymmetric

Skew-symmetricSkew-symmetric

Unitary matrixUnitary matrix

Given any matrix A with real entries:Given any matrix A with real entries:

symmetricskewisAA

symmetricisAAT

T

Complex conjugate of Complex conjugate of matricesmatrices

Alternative notation used in some books Alternative notation used in some books for Matrix Complex Conjugatefor Matrix Complex Conjugate

In this notes we use the bar *A

A

Complex HermitianComplex Hermitian

TH AA

TH AA

____

ii

i

i

iiH

521

643

526

143

Example:

Complex Hermitian Complex Hermitian PropertiesProperties

HHH

HH

HHH

HH

cc

ABAB

AA

BABA

AA

)(

definitionsdefinitions

examples:examples:

Hermitian:

Skew-Hermitian

Unitary

Given any matrix A with complex Given any matrix A with complex entries:entries:

MatrixhermitianSkewaisAA

MatrixHermitiananisAAT

T

(a) Find A such as:

10

653

02

1132 AAT

(b) Find A such as:

21

02

01

103 TAA

Exercises :

21

02

01

103 TAA

02

213 TAA

dc

baAGiven

02

21

33

33

db

ca

dc

ba

02

21

23

32

dbc

cba

021

21

21

A

ExercisesExercises

Ejercicio: Ejercicio: SimplificarSimplificar

TTT ABCX

TTT ABCX

TTTT ABCX

TTT ABCX

TBACX

Ejercicio: Ejercicio: SimplificarSimplificar TTT ABABY 1

TTTT BAABY 1

BAABYTT 1

BAABYTT 1

BAABYT1

BY 2