p1 rjm 06/08/02eg1c2 engineering maths: matrix algebra 1 eg1c2 engineering maths : matrix algebra dr...

p1 RJM 06/08/02 EG1C2 Engineering Maths: Matrix Algebra 1

EG1C2 Engineering Maths : Matrix AlgebraDr Richard Mitchell, Department of Cybernetics

Aim Describe matrices and their use in varied applicationsSyllabus

Introduction : why use; definitions & simple processingDeterminants and inversesTwo Port Networks, for electronics & other systemsGaussian elimination to solve linear equations + Gauss-JordanMatrix Rank and Cramer's Rule and TheoremEigenvalues and eigenvectors, applications incl. state spaceVectors - and their relationship with matrices.

ReferencesK.A.Stroud – Engineering Mathematics – Fifth Edn - PalgraveGlyn James - Modern Engineering Mathematics - Addison WesleyOnline Notes http://www.cyber.rdg.ac.uk/people/R.Mitchell/teach.htm


Matrix Algebra - Introduction

Simple systems - defined by equations y = f(x): e.g. y = kxMany systems involve many variables:

y1 = k11x1 + k12x2 + k13x3

y2 = k21x1 + k21x2 + k23x3

y3 = k31x1 + k32x2 + k33x3 etc.Matrix techniques allow us to represent these by y = k xBold letters show these are vector or matrix quantities.Why we use matricesCan group related data and process them together.Can use clever techniques to solve problems.Standard matrix manipulation techniques are available.Can use a computer to process the data: e.g. use MATLAB.In course use only 2 or 3 variables, use computers for more.


Example: Suspended Mass

Resolving forces in horizontal and vertical directions:cos (16.26) T1 = cos (36.87) T2

sin (16.26) T1 + sin (36.87) T2 = 300 {weight of mass}Simplifies to 0.96 T1 - 0.8 T2 = 0 and 0.28 T1 + 0.6 T2 = 300

16.26o 36.87o

300N

T2T1T1 & T2 are tensions in two wires.

(Angles chosen for easy arithmetic)

In Matrix Form:

2

1T

TT ,

0.60.28

8.00.96A and

300

0Y

The system can then be written as A.T = YT, Y are vectors - 1 column 2 rows, A is a matrix - 2 columns 2 rows


Example : Electronic Circuiti1 i3

i218

10 1512V

Using Kirchhoff’s Voltage and Current Laws First Loop 12 = 18 i1 + 10 i2 or 18 i1 + 10 i2 = 12 Second Loop 10 i2 = 15 i3 or -10 i2 + 15 i3 = 0 Summing currents i1 = i2 + i3 or -i1 + i2 + i3 = 0

Let

3

2

1

i

i

i

i

0

0

12

=v

111

15100

01018

=A then A.i = v


Matrix DefinitionsRectangular array of numbers, complex numbers, functions, .. If r rows & c columns, r*c elements, r*c is order of matrix.

mna...m2am1a

::2na...22a21a1na...12a11a

ija=A

m1a

..21a11a

1na....13a12a11a C R

6order matrix, 3*2 a is11042

135221

Square matrix: a11, a22, .. ann form the main diagonal

A vector has one column or one row

A is n * m matrixSquare if n = maij is in row i column j


Simple Matrix Operations

mna...m2am1a

::2na...22a21a1na...12a11a

=A

rcb...r2br1b

::2cb...22b21b1cb...12b11b

B

Equality : A and B are identical if, they are of the same size, m = r and n = c, andcorresponding elements are same ie aij = bij for all i,j

Illustrate these by defining A (size m*n) and B (size r*c)

43

21A

43

21B

43

11C

4

2 4321D A = B, but

A C, A D


Matrix Addition

mnbmna...m2bm2am1bm1a

::2nb2na...22b22a21b21a1nb1na...12b12a11b11a

=B+A=R

96

49

2742

1381

24

18

72

31 e.g.

42

31

25

14 Exercise

A and B must have same size: result is a matrix also of the same size, call it matrix R, in which for all elements, rij = aij + bij.

NB A + B = B + A (A + B) + C = A + (B + C) = A + B + C


Matrix SubtractionA and B must have same size: result is a matrix also of the same size, call it R, in which for all elements, rij = aij - bij.

52

27

2742

1381

24

18

72

31 e.g.

Matrix Scalar MultiplicationEach element in the matrix is multiplied by a scalar constant:

R = k.A Thus, each rij=k.aij.

301025

15510

6*52*55*5

3*51*52*5

625

312*5 e.g.

Note, k * (A + B) = k*A + k*B


Matrix Multiplication

R = A*B, number of columns in A = number of rows in B; R has number of rows as A and number of columns as B.e.g; if A is 2 * 3, B is 3 * 4 , then A * B is 2 * 4 matrix.

Do first element of ith row of A * first element of jth column of BMultiply second, third, etc. elements of these rows and columnsFind the sum of each product and store in rij

If A * B ok, then B * A is only possible if A & B are square.A * B B * A in general. A*(B*C) = (A*B)*C = A*B*CA*(B+C) = (A*B)+(A*C) (k*A)*B=k*(A*B)=A*(k*B) scalar k

n

1=kkjb*ikaijr :matrixresultant In


Examples & Exercise

27

17

1*36*4

1*56*2

1

6

34

52

5153

4117

1*11*40*12*13*41*11*10*42*1

1*31*10*22*33*11*21*30*12*2

121

130

012

*141

312

* *

* *

* *

* *

43

21

34

52


Multiplication and Example Systems

300

0

T 0.6 + T 0.28

T 0.8 -+ T 0.96

2T1T

*6.028.0

8.096.0

21

21

0

0

12

iii

i*15i*10-

i*10i*18

i

i

i

*

111

15100

01018

321

32

21

3

2

1

ExerciseExpress equations 5x + 2y = 16 and 3x = 18 – 4y in matrix form

y

x

Suspended Mass: A 2*2 matrix times a 2*1 vector = a 2*1 vector

Electronic Circuit: A 3*3 matrix times a 3*1 vector = a 3*1 vector


Matrix Transpose (A transposed is AT)

If R = AT, then rij = aji. If A is size m*n, then AT is size n*m.

mna...2na1na

:m2a...22a12am1a...21a11a

then

mna...m2am1a

::2na...22a21a1na...12a11a

TA A

42

31

43

21 T

63

52

41T

654

321 212

1 T

Note: (AT)T=A (A+B)T=AT+BT (A*B)T=BT*AT (kA)T=kAT

If AT=A, A is symmetrix matrix. If AT=-A, A is skew-symmetrix matrix

symmetric-skew is 02

20 symmetric is

34

41

p1 rjm 06/08/02eg1c2 engineering maths: matrix algebra 1 eg1c2 engineering maths : matrix algebra dr...

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