King M.R., Mody N.A. Numerical and Statistical Methods for Bioengineering: Applications in MATLAB
Подождите немного. Документ загружается.
A=
300
030
003
44
B = 2*ones(3,3) + diag([4 4 4])
B=
622
262
226
then,
44
C=B-A
C=
322
232
223
If we define
44
p = 0.5
44
D = p*B
we get the MATLAB output
p=
0.5000
D=
311
131
113
Transpose operations
The transpose of an m × n matrix A is an n × m matrix that contains the same elements as
that of A. However, the elements are “transposed” such that the rows of A are converted
into the corresponding columns, and vice versa. In other words, the elements of A are
“reflected” across the diagonal of the matrix. Symbolically the transpose of a matrix A is
written as A
T
.IfB = A
T
,thenb
ij
¼ a
ji
for all 1 ≤ i ≤ n and 1 ≤ j ≤ m.
For a 3 3 matrix C ¼
c
11
c
12
c
13
c
21
c
22
c
23
c
31
c
32
c
33
2
4
3
5
; its transpose is C
T
¼
c
11
c
21
c
31
c
12
c
22
c
32
c
13
c
23
c
33
2
4
3
5
:
Since a vector can also be viewed as a matrix, each element of a vector can be written
using double subscript notation, and thus the vector v can also be represented as
v ¼
v
11
v
21
:
:
:
v
n1
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
:
By taking the transpose of v we obtain
57
2.2 Fundamentals of linear algebra
v
T
¼½v
11
v
12
... v
1n
:
Thus, a column vector can be converted to a row vector by the transpose operation.
Using MATLAB
The colon operator can be used to represent a series of numbers. For example, if
44
a = [5:10]
then MATL AB creates a row vector a that has six elements as shown below:
a=
5678910
Each element in a is greater than the last by an increment of one. The increment can
be set to a different value by specifying a step value between the first and last values
of the series, i.e.
vector = [start value : increment : end value].
For example
44
b = [1:0.2:2]
b=
1.0000 1.2000 1.4000 1.6000 1.8000 2.0000
The transpose operation is performed by typing an apostrophe after the vector or
matrix varia ble. At the command prompt , type
44
v=a′
The MATLAB output is
v=
5
6
7
8
9
10
At the command prompt, type
44
v=b′
The MATLAB output is
v=
1.0000
1.2000
1.4000
1.6000
1.8000
2.0000
The transpose of the matrix A defined at the command prompt
44
A = [1 3 5; 2 4 6]
58
Systems of linear equations
A=
135
246
is equated to B:
44
B=A’
B=
12
34
56
Vector–vector multiplication: inner or dot product
The dot product is a scalar quantity obtained by a multiplication operation on two
vectors. When a column vector v is pre-multiplied by a row vector u
T
, and both
vectors u and v have the same number of elements, then the dot product (where · is
the dot product operator) is defined as follows:
u v ¼ u
T
v ¼ u
1
u
2
... u
n
½
v
1
v
2
:
:
:
v
n
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
¼
X
i¼n
i¼1
u
i
v
i
: (2:5)
It is straightforward to show that u
T
v ¼ v
T
u. Note that to obtain the dot product of
two vectors, the vector that is being multiplied on the left must be a row vector and the
vector being multiplied on the right must be a column vector. This is in accordance with
matrix multiplication rules (see the subsection on “Matrix–matrix multiplication” below);
i.e., if two matrices are multiplied, then their inner dimensions must agree. Here, a 1 × n
matrix is multiplied with an n × 1 matrix. The inner dimension in this case is “n.”
If a vector u is dotted with itself, we obtain
u u ¼ u
T
u ¼ u
1
u
2
... u
n
½
u
1
u
2
:
:
:
u
n
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
¼
X
i¼n
i¼1
u
2
i
: (2:6)
Using MATLAB
The scalar multiplication operator
*
also functions as the dot product operator. We define
44
u = [0:0.5:2.5]’
u=
0
0.5000
1.0000
1.5000
2.0000
2.5000
44
v = [0:2:10]’
59
2.2 Fundamentals of linear algebra
v=
0
2
4
6
8
10
Then, the following expressions are evaluated by MATLAB as
44
u’*v
ans =
55
44
v’*u
ans =
55
44
v’*v
ans =
220
44
u’*u
ans =
13.7500
Three-dimensional Euclidean space is a mathematical representation of our physical
three-dimensional space. Vectors in this space are commonly expressed in terms of
the mutually perpendicular unit (orthonormal) vectors i; j; and k. For example,
u ¼ u
1
i þ u
2
j þ u
3
k and v ¼ v
1
i þ v
2
j þ v
3
k, where u and v are expressed as a linear
combination of the orthonormal vectors i; j; k (see Section 2.2.4). The dot product
of two three-dimensional vectors u · v is calculated as
u v ¼ u
1
v
1
þ u
2
v
2
þ u
3
v
3
¼kukkvkcos θ;
where
kuk¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
2
1
þ u
2
2
þ u
2
3
q
¼
ffiffiffiffiffiffiffiffi
u
T
u
p
;
kvk¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
v
2
1
þ v
2
2
þ v
2
3
q
¼
ffiffiffiffiffiffiffi
v
T
v
p
;
and θ is the angle between the two vectors (kuk is the length of vector u; see
Section 2.2.3 on vector norms). If θ =908, then cos θ =0,andu and v are said to
be perpendicular or normal to each other.
Using MATLAB
If
44
a=[101];
44
b = [0; 1; 0];
44
v = a*b
MATLAB evaluates the dot product as
v=
0
60
Systems of linear equations
Since the dot product of a and b is zero, vectors a and b are said to be orthogonal
(or perpendicular in three-dimensional space) with respect to each other.
Matrix–vector multiplication
The outcome of a matrix–vector multiplication is another vector whose dimension
and magnitude depend on whether the vector is multiplied on the right-hand side or
the left-hand side of the matrix. The product of two matrices or a matrix and a vector
is usually not commutative, i.e. AB ≠ BA or Av ≠ vA. In this chapter, we will be
dealing only with the multiplication of a vector on the right-hand side of a matrix. A
matrix–vector multiplication, Av, involves a series of dot products between the rows
of the matrix A and the single column of vector v.Itisabsolutely essential that the
number of columns in A equal the number of rows in v, or, in other words, that the
number of elements in a row of A equal the number of elements contained in v.If
the elements of each row in A are viewed as components of a row vector, then A can
be shown to consist of a single column of row vectors:
A ¼
a
11
a
12
a
13
... a
1n
a
21
a
22
a
23
... a
2n
:
:
:
a
m1
a
m2
a
m3
... a
mn
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
¼
a
T
1
a
T
2
:
:
:
a
T
m
2
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
5
;
where a
T
i
¼ a
i1
a
i2
a
i3
... a
in
½and 1 i m.
If b is the product of the matrix–vector multiplication of A and v, then
b ¼ Av ¼
a
11
a
12
a
13
... a
1n
a
21
a
22
a
23
... a
2n
:
:
:
a
m1
a
m2
a
m3
... a
mn
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
v
1
v
2
:
:
:
v
n
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
¼
a
T
1
a
T
2
:
:
:
a
T
m
2
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
5
v
¼
a
T
1
v
a
T
2
v
:
:
:
a
T
m
v
2
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
5
¼
a
1
v
a
2
v
:
:
:
a
m
v
2
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
5
¼
a
11
v
1
a
12
v
2
a
13
v
3
... a
1n
v
n
a
21
v
1
a
22
v
2
a
23
v
3
... a
2n
v
n
:
:
:
a
m1
v
1
a
m2
v
2
a
m3
v
3
... a
mn
v
n
2
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
5
: (2:7)
Here, a
T
i
v ¼ a
i
v ¼ a
i1
v
1
þ a
i2
v
2
þþa
in
v
n
for 1 i m.
Matrix–matrix multiplication
The end result of the multiplication of two matrices is another matrix. Matrix multi-
plication of A with B is defined if A is an m × n matrix and B is an n × p matrix. The
resulting matrix C = AB has the dimension m × p. When two matrices are multiplied,
the row vectors of the left-hand side matrix undergo inner product or dot product
61
2.2 Fundamentals of linear algebra
vector multiplication
1
with the column vectors of the right-hand side matrix. An
element c
ij
in the resulting matrix C in row i and column j is the dot product of the
elements in the ith row of A and the elements in the jth column of B. Thus, the number
of elements in the ith row of A must equal the number of elements in the jth column of B
for the dot product operation to be valid. In other words, the number of columns in A
must equal the number of rows in B.Itisabsolutely essential that the inner dimensions
of the two matrices being multiplied are the same. This is illustrated in the following:
A B ¼ C
m nðÞn pðÞ¼ðm pÞ
_
same inner dimension
The matrix multiplication of A and B is defi ned as
C ¼ AB ¼
a
11
a
12
a
13
... a
1n
a
21
a
22
a
23
... a
2n
:
:
:
a
m1
a
m2
a
m3
... a
mn
2
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
5
b
11
b
12
b
13
... b
1p
b
21
b
22
b
23
... b
2p
:
:
:
b
n1
b
n2
b
n3
... b
np
2
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
5
¼
c
11
c
12
c
13
... c
1p
c
21
c
22
c
23
... c
2p
:
:
:
c
m1
c
m2
c
m3
... c
mp
2
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
5
;
(2:8)
where
c
ij
¼ a
T
i
b
j
or a
i
b
j
¼ a
i1
b
1j
þ a
i2
b
2j
þþa
in
b
nj
for 1 i m and 1 j p:
Note that because matrix–matrix multiplication is not commutative, the order of
the multiplication affects the result of the matrix multiplication operation.
A matrix can be interpreted as either a single column of row vectors (as discussed
above) or as a single row of column vectors as shown in the following:
B ¼
b
11
b
12
b
13
... b
1p
b
21
b
22
b
23
... b
2p
:
:
:
b
n1
b
n2
b
n3
... b
np
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
¼ b
1
b
2
... b
p
;
where
1
There is another type of vector product called the “outer product” which results when a column vector
on the left is multiplied with a row vector on the right.
62
Systems of linear equations
b
i
¼
b
1i
b
2i
:
:
:
b
ni
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
for 1 i p:
If A is represented by a column series of row vectors and B is represented by a row series
of column vectors, then the matrix multiplication of A and B can also be interpreted as
C ¼ AB ¼
a
11
a
12
a
13
... a
1n
a
21
a
22
a
23
... a
2n
:
:
:
a
m1
a
m2
a
m3
... a
mn
2
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
5
b
11
b
12
b
13
... b
1p
b
21
b
22
b
23
... b
2p
:
:
:
b
n1
b
n2
b
n3
... b
np
2
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
5
¼
a
T
1
a
T
2
:
:
:
a
T
m
2
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
5
b
1
b
2
...b
p
¼
a
T
1
b
1
a
T
1
b
2
... a
T
1
b
p
a
T
2
b
1
a
T
2
b
2
... a
T
2
b
p
:
:
:
a
T
m
b
1
a
T
m
b
2
... a
T
m
b
p
2
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
5
ðouter productÞ
Using MATLAB
The matrix multiplication operator is the same as the scalar multiplication operator,
∗. We define
44
A = [1 1; 2 2]; % 2 × 2 matrix
44
B = [1 2;1 2]; % 2 × 2 matrix
44
P = [3 2 3; 1 2 1]; % 2 × 3 matrix
Next we evaluate the matrix products:
44
C = A*B
C=
24
48
44
C = B*A
C=
55
55
44
C = A*P
C=
444
888
44
C = P*A
??? Error using ==> mtimes
Inner matrix dimensions must agree.
Can you determine why P ∗ A is undefined?
63
2.2 Fundamentals of linear algebra
2.2.3 Vector and matrix norms
A norm of a vector or matrix is a real number that provides a measure of its length or
magnitude. There are numerous instances in which we need to compare or determine the
length of a vector or the magnitude (norm) of a matrix. In this chapter, you will be
introduced to the concept of ill-conditioned problems in which the “condition number”
measures the extent of difficulty in obtaining an accurate solution. The condition number
depends on the norm of the associated matrix that defines the system coefficients.
The norm of a vector or matrix has several important propert ies as discussed in
the following.
(1) The norm of a vector v, denoted ||v||, and that of a matrix A,denoted||A||, is always
positive and is equal to zero only when all the elements of the vector or matrix are zero.
The double vertical lines on both sides of the vector that denote the scalar value of the
vector length should not be confused with the single vertical lines used to denote the absolute
value of a number. Only a zero vector or a zero matrix can have a norm equal to zero.
(2) The norm of the product of two vector s, a vector and a matrix, or two matrices is
equal to or less than the product of the norms of the two quantities being multiplied;
i.e., if u and v are vectors, and A and B are matrices, and the operations uv, Av, and
AB are all defined, then
(a) kuvkkukkvk,
(b) kAvkkAkkvk,
(c) kABkkAkkBk.
(3) For any two vectors or matrices, ku þ vkkukþkvk, which is also known as the
triangle inequality or the Cauchy–Schwarz inequa lity.
Norm of a vector
In Euclidean vector space,
2
the norm of an n × 1 vector v is defined as
kvk¼
ffiffiffiffiffiffiffi
v
T
v
p
¼
X
i¼n
i¼1
v
2
i
!
1=2
: (2:9)
The norm is thus the square root of the inner product of a vector with itself. This method
of calculating the vector length may be familiar to you since this is the same method used
to determine the length of vectors in our three-dimensional physical space.
A p-norm is a general method used to determine the vector length, and is defined as
kvk
p
¼
X
n
i¼1
jv
i
j
p
()
1=p
; (2:10)
where p can take on any value. When p = 2, we obtain Equation (2.9 ), which
calculates the Euclidean norm. Two other important norms commonly used are
the p = 1 norm and the p = ∞ norm :
p = 1 norm: kvk
1
¼ v
1
jj
þ v
2
jj
þ v
3
jj
þþ v
n
jj
;
p = ∞ norm: kvk
∞
¼ max
1in
jv
i
j.
2
Euclidean vector space is a finite real vector space and is represented as R
n
, i.e. each vector in this space is
defined by a sequence of n real numbers. In Euclidean space, the inner product is defined by Equation (2.5).
Other vector/function spaces may have different definitions for the vector norm and the inner product.
64
Systems of linear equations
Norm of a matrix
A matrix is constr ucted to perform a linear transformation ope ration on a vector.
The matrix operates or transforms the elements of the vector in a linear fashion to
produce another vector (e.g. by rotating and/or stretching or shrinking the vector). If
a vector of n elements is transformed into another vector of the same number of
elements, i.e. the size remains unchanged, then the operating matrix is square and is
called a linear operator.IfA is the matrix that linearly transforms a vector x into a
vector b, this linear transformation operation is written as
Ax ¼ b:
A matrix norm measures the maximum extent to which a matrix can lengthen a
unit vector (a vector whose length or norm is one). If x is a unit vector the matrix
norm is defined as
kAk¼max
kxk¼1
kAxk:
As we saw for vector norms, there are three commonly used matrix norms, the 1,
2, and ∞ norm. In this book we will concern ourselves with the 1 and ∞ norms, which
are calculated, respectively, as
kAk
1
¼ max
1jn
X
m
i¼1
ja
ij
j (2:11)
or the maximum sum of column elements, and
kAk
∞
¼ max
1in
X
n
j¼1
ja
ij
j (2:12)
or the maximum sum of row elements.
Using MATLAB
The norm function calculates the p-norm of a vector for any p ≥ 1, and the norm of a
matrix for p = 1, 2, and ∞. The syntax is
norm(x, p)
where x is a vector or matrix and p is the type of norm desired. If p is omitted, the
default value of p = 2 is used. We have
44
u=[111];
44
norm(u)
ans =
1.7321
44
norm (u,1)
ans =
3
44
norm(u,inf)
ans =
1
Note that infinity in MATLAB is represented by inf.
44
A=[32;45];
44
norm(A,1)
65
2.2 Fundamentals of linear algebra
ans =
7
44
norm(A, inf)
ans =
9
2.2.4 Linear combinations of vectors
A linear transformation operation is denoted by Ax = b; however, this equation is also
a compact notation used to represent a system of simultaneous linear equations. How
is this possible? If we assume that A is a 3 × 3matrixandx and b are 3 × 1 vectors, then
Ax ¼
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
2
4
3
5
x
1
x
2
x
3
2
4
3
5
¼
b
1
b
2
b
3
2
4
3
5
¼ b:
On performing the matrix–vector multiplication of Ax, and equating the inner
products between the three rows of A and the column vector x with the appropriate
element of b, we obtain
a
11
x
1
þ a
12
x
2
þ a
13
x
3
¼ b
1
;
a
21
x
1
þ a
22
x
2
þ a
23
x
3
¼ b
2
;
a
31
x
1
þ a
32
x
2
þ a
33
x
3
¼ b
3
:
If x ¼½x
1
x
2
x
3
is the only set of unknowns in this algebraic equation, then we
have a set of three simultaneous linear equations in three variables x
1
, x
2
, and x
3
. The
properties of the matrix A, which stores the con stant coefficients of the equations in a
systematic way, play a key role in determining the outcome and type of solution
available for a system of linear equations. A system of linear equations can have only
one of three outcomes:
(1) a unique solution ;
(2) infinitely many solutions;
(3) no solution.
The outcome of the solution solely depends on the properties of A and b. If a system
of linear equations yields either a unique solution or infinitely many solutions, then the
system is said to be consistent. On the other hand, if a system of linear equations has no
solution, then the set of equations is said to be inconsistent (See Problem 2.1).
The topics “linear independence of vectors” and “vector spaces” are crucial to the
understanding of the properties of A, and are discussed next.
If V ¼ðv
1
; v
2
; ...; v
n
Þis a set of vectors, and if another vector u can be expressed as
u ¼ k
1
v
1
þ k
2
v
2
þþk
n
v
n
;
where k
i
for 1 ≤ i ≤ n are scalars, then u is said to be a linear combination of the given
set of vectors V. For example, if
v
1
¼
2
3
; v
2
¼
1
1
;
and
u ¼
3
5
;
66
Systems of linear equations