ME 210 Applied Mathematics for Mechanical Enginee rs
Prof. Dr. Bülent E. Platin Spring 2014 – Sections 02 & 03 1/23
INTRODUCTION TO LINEAR ALGEBRA
Matrices and Vectors
ME 210 Applied Mathematics for Mechanical Enginee rs
Prof. Dr. Bülent E. Platin Spring 2014 – Sections 02 & 03 2/23
INTRODUCTION TO LINEAR ALGEBRA
Matrices and Vectors
Matrix: A rectangular array of scalars (numbers, variables, or functions – real or
complex).
Elements ; i = 1,2,...,m , j = 1,2,...,n
Rows
Columns
Element ai,j
i th row j th column
Size or dimension of a matrix: m x n
Total no of rows
Total no of columns
[ ] [ ]Aa
aaa
aaa
aaa
ij
mnm2m1
2n2221
1n1211
==
L
MOMM
L
L
ME 210 Applied Mathematics for Mechanical Enginee rs
Prof. Dr. Bülent E. Platin Spring 2014 – Sections 02 & 03 3/23
Vector: A matrix with only one row (size 1×n, row vector ) or only one column
(size m×1, column vector) .
Equality of matrices: Two matrices [A] and [B] are said to be equal to each other
if and only if
i) they have the same dimension m×n, and
ii) their corresponding elements are equal; i.e.,
aij = bij for all i = 1, 2, …, m and j = 1, 2, …, n
Addition/Subtraction of matrices: Addition/subtraction is defined only for matrices
of the same size and result in another matrix of the same size.
For two matrices [A] and [B] of the same size [C] = [A] ± [B] implies that
cij = aij ± bij for all i = 1, 2, …, m and j = 1, 2, …, n
ME 210 Applied Mathematics for Mechanical Enginee rs
Prof. Dr. Bülent E. Platin Spring 2014 – Sections 02 & 03 4/23
Example:
Given
4 6 3 5 1 0 4 5 6 1 3 0 1 5 3[ ] [ ] = + = =
0 1 2 3 1 0 0 3 1 1 2 0 3 2 2A B
− − − + − + + + + +
Multiplication/Division by a scalar: Multiplication/division of a matrix by a scalar k
implies that all its elements are to be multiplied/divided by the same scalar k; i.e.,
k [A] = [k aij] or [A] / k = [aij /k]
4 6 3 5 1 0 4 5 6 1 3 0 9 7 3[ ] - [ ] = + = =
0 1 2 3 1 0 0 3 1 1 2 0 3 0 2A B
− − − − + − − − − − −
–
−=
−=
013
015[B]and
210
364[A]
ME 210 Applied Mathematics for Mechanical Enginee rs
Prof. Dr. Bülent E. Platin Spring 2014 – Sections 02 & 03 5/23
Example:
Given2.7 -1.8
[A]= 0.9 3.6
Then
Some important properties of matrices: Given matrices [A], [B], and [C] of the
same size and a set of scalar constants k1, k2, and k3, the following properties hold:
[A] + [B] = [B] + [A] commutative
[A] + ([B] + [C]) = ([A] + [B]) + [C]) associative
k1 ([A] + [B]) = k1 [A] + k1 [B] distributive
(k1 + k2) [A] = k1 [A] + k2 [A]
k1 (k2 [A]) = k2 (k1 [A]) = (k1k2) [A]
−=
−=
41
23[A]
910
and7.21.8
3.65.4[A]2
ME 210 Applied Mathematics for Mechanical Enginee rs
Prof. Dr. Bülent E. Platin Spring 2014 – Sections 02 & 03 6/23
=[B]
The elements of [C] are given as cij = ai1 b1j + ai2 b2j + ….. + ain bnj =
Matrix multiplication: Let [A] be a m×n matrix and [B] be a p×q matrix. The matrix
product [A] [B] is defined only if n = p and it gives a matrix [C] of size m×q and
shown as [C] = [A] [B]
Note that the element cij can be
interpreted as the dot product
(inner product) of the
ith row vector of [A]
and the
jth column vector of [B].
∑=
n
1kkjikba
=[A]
ME 210 Applied Mathematics for Mechanical Enginee rs
Prof. Dr. Bülent E. Platin Spring 2014 – Sections 02 & 03 7/23
1 0 23 2 1 7 2 6
[ ] [ ][ ] 5 3 10 4 6 56 36 16
6 4 2
C A B
− = = =
Example:
Given
Since number of columns of [A] and number of rows of [B] match, [A] and [B] matrices are called compatible as far as [A] [B] operation is concerned.
Hence, their product becomes
=
−=
246
135
201
[B]and640
123[A]
ME 210 Applied Mathematics for Mechanical Enginee rs
Prof. Dr. Bülent E. Platin Spring 2014 – Sections 02 & 03 8/23
Note that, in this example, even though [A][B] multiplication is defined, [B][A]multiplication is not defined since the number of columns of [B] (which is 3) andnumber of rows of [A] (which is 2) do not match.
Matrix multiplication is not commutative .
Even in cases where both [A][B] and [B][A] multiplications are defined, these multiplications are usually not equal to each other; i.e.,
[A] [B] ≠ [B] [A] , in general
Note that only if the matrix [A] is of size m×n and [B] is n×m, then both [C] = [A][B] and [D] = [B][A] are defined where [C] and [D] are square matrices of different sizes m and n, respectively; therefore, not equal to each other.
If matrices [A] and [B] are both square and of the same size n, then both [C] = [A][B] and [D] = [B][A] are defined where [C] and [D] are square matrices of the same size n; but not necessarily equal to each other.
ME 210 Applied Mathematics for Mechanical Enginee rs
Prof. Dr. Bülent E. Platin Spring 2014 – Sections 02 & 03 9/23
Example:
Transpose of a matrix: The transpose of an m×n matrix [A] is an n×m matrix [B]
whose rows are the columns and columns are the rows of [A]. It is denoted by [A]T.
Hence, [aij]T = [aji]
Given square matrices
−=
=
02
12[B]and
30
21[A]
−−
=
−
=
06
12
02
12
30
21[A][B]
−−=
−=
42
72
30
21
02
12[B][A]
Example:
Given square matrices
=
=
20
12[B]and
10
11[A]
=
=
20
32
20
12
10
11[A][B]
=
=
20
32
10
11
20
12[B][A]
≠≠≠≠
=
ME 210 Applied Mathematics for Mechanical Enginee rs
Prof. Dr. Bülent E. Platin Spring 2014 – Sections 02 & 03 10/23
Example:
Note that [b]T is a convenient way to describe a row vector.
The following are two important properties of transpose operation.
([A] + [B])T = [A]T + [B]T
([A] [B])T = [B]T [A]T Note the change of the order
−=→
−=
01
08
45
[A]004
185[A] T [ ]257[b]
2
5
7
[b] T =→
=
ME 210 Applied Mathematics for Mechanical Enginee rs
Prof. Dr. Bülent E. Platin Spring 2014 – Sections 02 & 03 11/23
Examples of sparse matrices:
Square matrix: A matrix with equal number of rows and columns.
Principal (main) diagonal
Off-diagonal elements
=
−=
=
000
000
002
010
000
[C]
0050
0001
0000
0300
[B]001
200[A]
Sparse matrix: A matrix with most of its elements zero
[ ]
=
nnn2n1
2n2221
1n1211
aaa
aaa
aaa
A
L
MOMM
L
L
ME 210 Applied Mathematics for Mechanical Enginee rs
Prof. Dr. Bülent E. Platin Spring 2014 – Sections 02 & 03 12/23
Symmetric matrix: A square matrix whose off–
diagonal elements at symmetric locations are equal;
that is aij = aji for all i, j (=1, 2, …, n)
Note that the transpose of a symmetric matrix [A] is
equal to itself; i.e., [A]T = [A]
Skew-symmetric matrix: A square matrix whose
off–diagonal elements at symmetric locations are
equal in size but of opposite sign; that is
aij = –aji for all i, j (=1, 2, …, n)
Note that the elements of main diagonal of a
skew-symmetric [A] are all zero; i.e.,
aii = 0 for all i =1, 2, …, n
Example:
−−
−=
425
201
513
[A]
Example:
−−
−=
051
504
140
[B]
ME 210 Applied Mathematics for Mechanical Enginee rs
Prof. Dr. Bülent E. Platin Spring 2014 – Sections 02 & 03 13/23
Note that any square matrix, [A], can be written as [A] = [A]s + [A]sk
where
is called the symmetric part of [A], and
is called the skew-symmetric part of [A].
2[A][A]
[A]T
s+=
2[A][A]
[A]T
sk−=
Example:
−−
−=
437
101
313
[A]
−−
−=
−−
−+
−−
−=
425
201
513
413
301
713
437
101
313
21
[A]s
−
−=
−−
−−
−−
−=
012
100
200
413
301
713
437
101
313
21
[A]sk
ME 210 Applied Mathematics for Mechanical Enginee rs
Prof. Dr. Bülent E. Platin Spring 2014 – Sections 02 & 03 14/23
Triangular matrix: A square matrix whose off–diagonal elements above or below
the main diagonal are all zero.
Example: Upper triangular matrix
Lower triangular matrix
Diagonal matrix: A square matrix whose off–diagonal elements are all zero.
That is aij = 0 for all i , j = 1, 2, …, n, i ≠ j
−−
=400
210
513
[A]
−
−=
451
014
003
[B]
−=
400
010
003
[A]
ME 210 Applied Mathematics for Mechanical Enginee rs
Prof. Dr. Bülent E. Platin Spring 2014 – Sections 02 & 03 15/23
Identity (unity) matrix: A diagonal matrix whose elements in the principal diagonal
are all 1. That is aii = 1 for all i = 1, 2, …, n
Null matrix: A m×n matrix whose elements are all zero. That is,
aij = 0 for all i = 1, 2, …, m and j = 1, 2, …, n
=
=
1000
0100
0010
0001
[B]10
01[A]
=
=
=00000
00000[C]
0000
0000
0000
0000
[B]
00
00
00
[A]
ME 210 Applied Mathematics for Mechanical Enginee rs
Prof. Dr. Bülent E. Platin Spring 2014 – Sections 02 & 03 16/23
Note that, contrary to the case in multiplications of scalars, if the multiplication of
two square matrices gives a null matrix then this does not imply that at least one
of the matrices multiplied should be a null matrix as well.
Banded matrix: A square matrix; some of its diagonals next to the main diagonal
are not zero, but the rest of the off-diagonal elements are all zero.
−−=
=→
−−
=
=
11
11[B][A]but
00
00[A][B]
11
11[B]&
22
11[A]
ME 210 Applied Mathematics for Mechanical Enginee rs
Prof. Dr. Bülent E. Platin Spring 2014 – Sections 02 & 03 17/23
Example:
Consider the free (unforced) motion of the following model which may be representing
the longitudinal vibrations of a long elastic body whose both ends are fixed.
Using the Newton’s 2nd law of motion for each mass, the governing equations of motion
can be written as:
k1 k2 k3 k4 k5 k6
m1 m2 m3 m4 m5
x1 x2 x3 x4 x5
0xk)x(xkxm
0)x(xk)x(xkxm
0)x(xk)x(xkxm
0)x(xk)x(xkxm
0)x(xkxkxm
5645555
54534444
43423333
32312222
2121111
=+−+=−+−+=−+−+=−+−+
=−++
&&
&&
&&
&&
&&
ME 210 Applied Mathematics for Mechanical Enginee rs
Prof. Dr. Bülent E. Platin Spring 2014 – Sections 02 & 03 18/23
[ ] [ ] + [ ] [ ] = [0]M x K x&&
Mass matrixdiagonal Position vector
Stiffness matrix, symmetrical and banded
Forcing vectornull vector
Governing equation in matrix form:
=
5
4
3
2
1
m0000
0m000
00m00
000m0
0000m
[M]
=
5
4
3
2
1
x
x
x
x
x
[x]
+−−+−
−+−−+−
−+
=
655
5544
4433
3322
221
kkk000
kkkk00
0kkkk0
00kkkk
000kkk
[K]
=
0
0
0
0
0
[0]
ME 210 Applied Mathematics for Mechanical Enginee rs
Prof. Dr. Bülent E. Platin Spring 2014 – Sections 02 & 03 19/23
Determinants, Minors, and Cofactors
Determinant:
It is a scalar quantity and defined only for square arrays.
Every square array A of size n has a unique determinant value D and it is shown as
[A] detA
aaa
aaa
aaa
D
nnn2n1
2n2221
1n1211
===
L
MOMM
L
L
ME 210 Applied Mathematics for Mechanical Enginee rs
Prof. Dr. Bülent E. Platin Spring 2014 – Sections 02 & 03 20/23
Minor:
It is a determinant of an array, which is one lower size than the array of an original
determinant.
A minor is always associated with only one of the elements of an original array.
Its array is obtained by deleting the row and column of the original array containing
that particular element; that is the minor Mij of the element aij of an array of size n is
the determinant of the array of size n–1, which is formed by deleting the ith row and
jth column of the original array.
Example:
425
201
513
A
−−
−=
42
20M11 −
−=→
21
53M32 −
−=
ME 210 Applied Mathematics for Mechanical Enginee rs
Prof. Dr. Bülent E. Platin Spring 2014 – Sections 02 & 03 21/23
Cofactor: It is also associated with only one of the elements of an original array and defined as Cij = (–1)i+j Mij
Example:
Evaluation of Determinants:
The value of the determinant of size one is same as the scalar involved; that is aa =
The value of the determinant of size two is obtained as
211222112221
1211 aaaaaa
aa−=
425
201
513
A
−−
−=
42
20M1)(C 11
1111 −
−=−=→ +
21
53M1)(C 32
2332 −
−−=−= +
ME 210 Applied Mathematics for Mechanical Enginee rs
Prof. Dr. Bülent E. Platin Spring 2014 – Sections 02 & 03 22/23
Example:
exp . . . 1
1 3 06 4 2 4 2 6
D 2 6 4 1 3 0 (12 0) 3(4 4) 120 2 1 2 1 0
1 0 2stansion w r t row
= = − + = − − + = −− −
− 144444424444443
Determinants of size three and higher can be evaluated by expanding it with
respect to one of its rows or columns as
∑∑==
===n
jany1i
ijij
n
iany1j
ijij
nnn2n1
2n2221
1n1211
CaCa
aaa
aaa
aaa
D
L
MOMM
L
L
ME 210 Applied Mathematics for Mechanical Enginee rs
Prof. Dr. Bülent E. Platin Spring 2014 – Sections 02 & 03 23/23
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