MECH0801-wk-02-lec

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Dr.Thitaphol Huyanan

MECH-0851FINITE ELEMENTANALYSISPreliminary Basics I

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ClassDr.Thitaphol Huyanan

Agenda1

Basic Matrices

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Basic Matrices2

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Class

Basic Definitions3

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Matrices4

A matrixis an array of ordered numbers

which can be represented by a single symbol.

The matrix consists ofmnmembers

arrangedinmrowsandncolumns.

11 12 1 1

21 22 2 2

1 2 ( )

1 2

1 2

12

j n

j n

iji i ij in m n

m m mj mn

j n

i

m

a a a aa a a a

a Aa a a a

a a a a

A Aread A bar

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Vectors and Scalars

A vectoris a matrix with only one column (n =

1), it is also called as a column vector.

A scalaris a matrix which has one row(m = 1)and one column (n = 1).

1 1

2 2

( 1)i

m

m m

u u

u uu u

u u

u u

5

read u tilde

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Special Matrices6

A transpose of a matrix is another

matrix which is

created by writing the rows of as the columns

of andwriting the columns of as the rows of byreflecting on its main diagonal.

A TA

ATA

TAA A

11 12 1

21 22 2

( )

1 2 11 2

trans

1 1

12 22p 2

( )

1

e

2

os

n

n

m n

m m mn m

TTm

ij ij jin m

n n mn

a a aa a a

a a a a a aa a a

a a a

a a a

A

A

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Special Matrices (continued)7

1

21 2

1 2

(1 )

T

n j

nn

n

T

b

bb b b b

b b b b b

b Tb

A transpose of column vector gives a matrix with

one row ( m = 1) which is so-called as a row vector.

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Special Matrices (continued)8

A matrix of any order whose all members arezero is referred to as a null matrixand isdesignated as

A matrix that consists of the same number ofrows and columns (nn) is called a square matrix.

11 12 1 1

21 22 2 2

1 2 ( )

1 2

j n

j n

iji i ij in n n

n n nj nn

a a a a

a a a aa A

a a a a

a a a a

A A

( )

0m n

0 0

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Special Matrices (continued)

A square matrix

whose memberaij=0

when ij is a diagonal

matrix.

9

A square matrix

whose memberaij=0

when ij while aii=1

is known as a unit(oridentity) matrix.

11

22

33

0 0 0

0 0 0

0 0 0

0 0 0 nn

d

d

d

d

D

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

I

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Special Matrices (continued)

A square matrix

whose memberaij=0

when i>j is an upper

triangular matrix.

10

A square matrix

whose memberaij=0

when i

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Class

Matrix Algebra11

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Equality of Matrices12

Matrices and are equalif, and only if,

has the same dimension as and aij=bij for all

members.

A B AB

1 2 1 3 1 2 1 32 3 4 1 2 3 4 1

1 4 0 2 1 4 0 2

3 1 2 3 3 1 2 3

andA B

A B

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Addition and Subtraction13

11 12 1 11 12 1

21 22 2 21 22 2

1 2 1 2

11 11 12 12 1 1

21 11 22 22 2 2

1 1 2 2

n n

n n

m m mn m m mn

n n

n nij

m m m m mn mn

a a a b b b

a a a b b b

a a a b b b

a b a b a ba b a b a b

r a

a b a b a b

A B

R A B

and

ij ijb

Matrices and can be addedorsubtracted

if, and only if, their dimensions are the same.

A B

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Multiplication14

Scalar Multiplication,

Dot Productof vectors, it is a scalar sum ofproducts. Thus the dot product of ordernth vectorsand can then be obtained as,

11 12 1

21 22 2

1 2

n

nij

m m mn

a a aa a a

a

a a a

A

1 1

1

1

( )

( )

( )

1 1 2 21

2

1 2

1

n

n

n n

nT

n i i

i

n

r x y x y x y yy

x x x r x y

y

x y

x yr

No. of columns of

multiplicand and no. ofrows of multiplier must

x

y

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Multiplication (continued)15

Matrix Multiplication, let consider matrices andas the vector of row vector components and therow vector of vector components, respectively.

A B

11 12 1 11 12 1

21 22 2 21 22 2

1 2 1 2

11 12 1 11 12

21 22 2 21

11 2

1

2

q q

q q

m m mq m m mq

n

n

qq q qn

T

T

m qTm

q n

a a a a a a

a a a a a a

a a a a a a

b b bb bb b b b

bb b b

aaA

a

B1

22 2

2

1 2

th

n

n

q qn

n

T

b

b b

b b

q

b b b

a bwhere and are the order vectors.

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Multiplication (continued)16

Soaproductofmatrixmultiplicationisobtainedas

1

21 2

( ) ( )

1 1 1 2 1 11 12 1

21 22 22 1 2 2 2

1 21 2

nm q q n

m

n n

nn

m m mnm m m n

r r r

r r r

r r r

T

T

T

T T T

T T T

T T T

aaA B b b b

a

a b a b a b

a b a b a b

a b a b a b

( ) 1

q

ij ij i j ik kj

m n k

r r a b

TR a b, where

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Transpose of Products of

Matrices17

Given matrices , and a scalar , then

A B

TT

A A

T TA A

T

T T TA B A B

T T T

A B B A

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Class

Determinants18

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Determinant of Matrix19

Every square matrix is associated with a

single particular scalar quantity called the

determinantof , denoted by or .

AA

A det A

Determinant of Order One,11 11

(1 1)

det a a

A A A

11 1211 22 12 21

(2 2) 21 22

deta a

a a a aa a

A A ADeterminant of Order Two,

Determinant of Order Three,

Rule of Sarrus (+) (+) (+)

() () ()

11 12 13 11 12 13 11 22 33 11 23 3211 12

21 22 23 21 22 23 21 22 12 23 31 12 21 33(3 3)

31 3231 32 33 31 32 33 13 32 21 13 31 22

a a a a a a a a a a a aa aa a a a a a a a a a a a a a

a aa a a a a a a a a a a a

A A

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Determinant of Matrix

(continued)20

Consider the determinant of ordern2 of a n

n matrix ,

11 1211 22 12 21 11 22 12 21

21 22(2 2)

( ) ( )a a

a a a a a a a aa a

A A

A

If the determinant of ordern1 that obtained

by eliminating the row i and the columnj in the

matrix is defined as a minorof the element in rowi and columnj, denoted byMij , then a22 and a21

becameM11 andM12 respectively. Thus,

A

11 22 12 21 11 11 12 12

a a a a a M a M A

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Determinant of Matrix

(continued)21

Similar to aforementioned, the determinant ofcan also be obtained as followings.

11 1211 22 12 21 21 12 22 11 21 21 22 22

21 2211 22 21 12 11 11 21 21

12 21 22 11 12 12 22 22

a aa a a a a a a a a M a M

a a

a a a a a M a M a a a a a M a M

A

A

Therefore,1 1 1 2

11 11 12 12 11 11 12 12 1 12 1 2 2

21 21 22 22 21 21 22 22 2 21 1 2 1

11 11 21 21 11 11 21 21 1 11 2 2 2

12 12 22 22 12 12 22 22 2

1 1

1 11 1

1 1

( ) ( )

( ) ( )( ) ( )

( ) ( )

j j

j j

i i

i

a M a M a C a C a C

a M a M a C a C a C a M a M a C a C a C

a M a M a C a C a C

A

2i

Where Cij1ij Mij is a cofactorof the

element in rowiand column

j.

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Determinant of Matrix (continued)22

Hence, a determinant of any nth order matrix

can be obtained by multiplyingALL the elements in

ANYrow (orANYcolumn) by their own cofactors

and addingALL the resulting products.th

( ) 1

th

1

det

( 1)

n

j jn n j

n

i i

i

i j

i j

i j i j

a C i

a C j

C i j

C M

i i

j j

A A A

where

, along the row ,

, along the column

or

cofactor of the element in row and column

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Adjoint of Matrix23

Adjoint of matrix is a transpose of a matrix

of cofactors which has a same dimension as and

it can be expressed as11 21 1 1

12 22 2 2

( ) 1 2

1 2

adjT

n n

i n

i n

ijj j ij nj

n n in nn

C C C C

C C C C

CC C C C

C C C C

TA A C

AA

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Questions and Discussion24

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MECH0801-wk-02-lec