Transcript Scripts and Functions

Vectors and Matrices
Learning
Objectives
Understand the
nature of matrices
Understand how to
manipulate
matrices in Matlab
AE6382 Design Computing
Lecture
• Basic vector & matrix
concepts
• Creating arrays and matrices
• Accessing matrix
components
• Manipulating matrices
• Matrix functions
• Solving simultaneous
equations
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Fall 2006
Using Matlab with Arrays and Matrices
• Matlab’s origins are in the early efforts to develop fast
and efficient programs for handling linear equations…
– Operations with arrays, vectors and matrices are needed
– Only the most computationally efficient routines are used
– Matlab is very “C-like” but adds a number of operators and
extends its syntax to handle a range of array, vector and matrix
operations
– Matlab’s fundamental data structure is the array and vectors
and matrices follow easily
– BUT… to see some of the power of Matlab for engineering
applications, we’ll have to dig a bit more deeply into some of
the underlying math (no, this is not going to turn into a math
class, but it’s often hard to avoid math in engineering)
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Basic Concepts
Scalars: magnitude only
x, mass, color, 13.451
Vectors: magnitude AND direction
force  12.74 i  5.234 j
H  a1i1  a2i2  a3i3  a4i4
 anin
Arrays: can be 2D or higher dimension
 a11 a12 a13 a14 
A   a21 a22 a23 a24 
 a31 a32 a33 a34 
R   rijk 
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Matlab Can Handle This…
Scalars:
>> whos
Name
a
density
mass
resistance
s
stress
Size
1x1
1x1
1x1
1x1
1x1
1x1
Bytes
8
8
8
8
8
8
Class
double
double
double
double
double
double
array
array
array
array
array
array
Vectors:
>> force=[12.3, 5.67]
force =
12.3000
5.6700
>> hvec=[1, 5, -3, 4, 0]
hvec =
1
5
-3
4
0
Arrays:
>> coef=[1, 2; -4, 3]
coef =
1
2
-4
3
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Fall 2006
Basic Array Operations
• Addition/subtraction: C=A+B where cij = aij+bij
>> C=A+B
B =
A =
1
2
-4
1
-4
3
-3
6
C =
-3
3
-7
9
• Multiplication/division: C=A .* B where cij = aij*bij
>> C=A.*B
>> C=A./B
C =
C =
-4
2
-0.2500
2.0000
12
18
1.3333
0.5000
• Exponentiation: C=A .^ 4 where cij = aij4
>> C=A.^2
C =
1
4
16
9
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Fall 2006
Notes on Array Operations
• Arithmetic operations on arrays are just like the same
operations for scalars but they are carried out on an
element-by-element basis.
– the dot (.) before the operator indicates an array operator; it is
needed only if the meaning cannot be automatically inferred.
>> A=[1 2 3 4 5];
>> 2.*A
ans =
2
4
6
8
10
>> 2*A
ans =
2
4
6
8
10
>> B=[2 4 6 8 10];
>> A.*B
ans =
2
8
18
32
50
>> A*B
??? Error using ==> *
Inner matrix dimensions must agree.
– when combining arrays, make sure they all have the same
dimensions
– applies to vectors, 2D arrays, multi-dimensional arrays
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Fall 2006
More Notes on Array Operations
• Most Matlab functions will work equally well with both
scalars and arrays (of any dimension)
>> A=[1 2 3 4 5];
>> sin(A)
ans =
0.8415
0.9093
0.1411
-0.7568
-0.9589
1.4142
1.7321
2.0000
2.2361
>> sqrt(A)
ans =
1.0000
• Use brackets […] to construct arrays
• Use colon notation (e.g., A(:,2) or f(3:11) to index)
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Fall 2006
Array Constructors
• Arrays are often read into Matlab from files or entered
by the user…
• But building arrays from scratch can be tedious
– Explicit:
>> g(1)=1; g(2)=3; g(3)=-4
g =
1
3
-4
– Using Matlab array constructors:
>> A=ones(2,3)
A =
1
1
1
1
1
1
>> B=-3*ones(1,5)
B =
-3
-3
-3
-3
-3
>> C=zeros(2,3)
C =
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0
0
0
0
0
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Fall 2006
Let’s Build Some Arrays…
What will these produce?
>> A=3*eye(2,2)
A =
3
0
0
3
D = magic(5)
>> B=diag([1 2 3 4])
diag(D)
B =
diag(diag(D))
1
0
0
0
0
2
0
0
0
0
3
0
0
0
0
4
>> C=diag([1 2 1],1)
Z = [magic(3),zeros(3,2), -ones(3,1);
4*ones(2,4), eye(2,2)]
Z(:,3)=[]
mess = 10*rand(4,5)
messy = 10*randn(4,5)
C =
0
1
0
0
0
0
2
0
0
0
0
1
0
0
0
0
test = 1./(3*ones(2,3)
>> diag(A)
ans =
3
3
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Fall 2006
Vectors and Matrices
• We’ve referred to vectors and matrices frequently… but
exactly what are we talking about?
– what is a matrix?
– is it different from an array?
• ANSWER:
– vectors and matrices are arrays with an “attitude”
– that is, they look just like an array (and they are arrays), but
they live by a very different set of rules!
– Vectors:
f b  ?
3f ?
f g ?
s r  ?
Can you explain what, if
anything, results from these
operations with vectors?
h ?
a /b  ?
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Fall 2006
Why Matrices?
• A matrix is an array that obeys a different set of rules
– addition & subtraction are same as for arrays,
– but multiplication, division, etc. are DIFFERENT!
– a matrix can be of any dimension but 2D square matrices are
the most common by far
• A large and very useful area of mathematics deals with
what is called “linear algebra” and matrices are an
integral part of this.
• Many advanced computational methods in engineering
make extensive use of linear algebra, and hence of
matrices
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A Simple Example
• A set of simultaneous linear algebraic equations will
often arise in engineering applications
3x  2 y  14
x  4 y  14
• How do you solve these?
– Solve first for x in terms of y; substitute in second and solve for
y; use this in first to find x
– Use “Cramer’s Rule”
– Other?
• Let’s try a more abstract notation:
3 2   x   14 
OR
1 4   y   14

  

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C*z = b
Fall 2006
A Simple Example-cont’d
• What do we mean by the * for this form?
3 2  x  3x  2 y 
C* z  
*   


1 4   y   x  4 y 
– Note that the column matrix, z, is multiplied times the first row
of C on an element-by-element basis and the results are
summed to get the first row of the answer
– Ditto for the second row…
– This is NOT array multiplication; it is matrix multiplication
• For two 2D matrices in general:
A*B = C
where :
NOTE: the number of
columns in A must be equal
to the number of rows in B
(N in this example)
N
cij   aik bkj
k 1
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Fall 2006
A Few Notes on Matrices
• Matlab handles matrix multiplication with the * symbol (NOTE: this
is NOT array multiplication!)
– From our formula we see that in general: A*B  B*A
– In other words, matrix multiplication is NOT commutative
• Matrices behave just like arrays for addition and subtraction
• Matrix division is not strictly defined but a matrix inverse is
available to address this situation, among others.
–
–
–
–
suppose: 3y=6 and you need to find y…
The usual approach: y=6/3=2 (division by 3)
Also useful: y=3-1*6=2 (multiplication by the inverse of 3)
If we don’t know how to divide, we can accomplish the same by using
the notion of the inverse. Recall definition of inverse:
(value) *(inverse value)  1
– Turns out we know how to compute matrix inverses (but it requires a
lot of computational effort)
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Fall 2006
Let’s Solve Our Problem Using Matlab
>> coef=[3 -2; 1 4]
coef =
3
-2
1
4
>> inv(coef)
% Matlab has the inv() function
ans =
0.2857
0.1429
-0.0714
0.2143
>> b=[14 -14]'
b =
14
-14
>> z=inv(coef)*b
z =
2
-4
>> coef*z
% Let's check our answer!
ans =
14
-14
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Fall 2006
Some More Notes:
• Using the Matlab inv() function is not always best
– It can take a VERY long time for large matrices
– The inverse may have poor precision for some kinds of matrices
• If you just want to solve the set of equations, there are
much quicker and more accurate methods
– Uses powerful algorithms from linear algebra
– Notation is tricky because it introduces the concept of a “left” and
a “right” matrix division in Matlab
Given :
C* z = b
Left  divide both sides by C :
C \ C* z = C \ b
or :
z =C\b
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NOTE:
C\C=1, and
1*anything=anything
Fall 2006
Let’s Try This Out…
coef =
3
-2
1
4
>> b
b =
14
-14
>> zz=coef\b
zz =
2.0000
-4.0000
OK, now what do you think these expressions yield?
coef\eye(2,2)
coef\eye(2,2)*coef
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Things Can Get Weird…
• We usually think of the unknown (z) as a column matrix
and the RHS (b) as a column matrix also
• In some fields, it is more useful if these are ROW
matrices
– One formulation can easily be converted into the other!
– We can treat either formulation in Matlab
• First, ON YOUR OWN, prove from our multiplication
formula that:
( A * B)T  BT * AT
• Now, using this, we take the transpose of our equation:
T
T
T
z
  x y
(C * z )  b
where
T
T
T
T
b
 14 14
z *C  b
 3 1
C 


2
4


T
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Fall 2006
Let’s Try It Out in Matlab:
>> coefT=coef'
coefT =
3
1
-2
4
>> bT=b'
bT =
14
-14
>> zT=bT*inv(coefT)
zT =
2
-4
>> % ALSO WE CAN USE RIGHT DIVIDE:
>> zT2=bT/coefT
zT2 =
2.0000
AE6382 Design Computing
-4.0000
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Fall 2006
Other Matlab Matrix Functions
• So far we’ve only scratched the surface of Matlab’s
abilities to work with matrices…
• Matrices can contain COMPLEX numbers
• Some of the other matrix functions are:
–
–
–
–
–
–
det(A): determinant of the matrix
rank(A): rank of the matrix
trace(A): sum of diagonal terms
sqrtm(A): matrix square root (i.e., sqrtm(A)*sqrtm(A)=A)
norm(A): matrix norm (useful for vector magnitudes)
eig(A): eigenvalues and eigenvectors of matrix
– …
• Keep in mind that Matlab is using some of the latest
and most powerful algorithms to compute these
functions.
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Fall 2006
Finally, What About Vectors?
• The matrix and array operations and functions can be
used to manipulate vectors, but you’ll have to be careful
• Vector dot product:
>> f=[1 2]'
f =
Column vectors
>> f=[1 2]
f =
1
2
1
Row vectors
>> g=[4 -3]'
2
>> g=[4 -3]
g =
g =
4
4
-3
>> fdotg=f*g'
-3
fdotg =
>> fdotg=f'*g
-2
fdotg =
-2
>> gdotf=g*f'
gdotf =
• ON YOUR OWN:
-2
– Vector magnitude?
– Vector cross product?
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