Vectors and Matrices

1
Vectors and Matrices

• Lecture
• Basic vector matrix concepts
• Creating arrays and matrices
• Accessing matrix components
• Manipulating matrices
• Matrix functions
• Solving simultaneous equations

Learning Objectives Understand the nature of
matrices Understand how to manipulate matrices in
Matlab
2
Using Matlab with Arrays and Matrices

• Matlabs 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
• Matlabs fundamental data structure is the array
  and vectors and matrices follow easily
• BUT to see some of the power of Matlab for
  engineering applications, well 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
  its often hard to avoid math in engineering)

3
Basic Concepts
Scalars magnitude only
x, mass, color, 13.451
Vectors magnitude AND direction
Arrays can be 2D or higher dimension
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Matlab Can Handle This
Scalars
gtgt whos Name Size Bytes
Class a 1x1 8
double array density 1x1
8 double array mass 1x1
8 double array resistance 1x1
8 double array s 1x1
8 double array stress 1x1
8 double array
Vectors
gtgt force12.3, 5.67 force 12.3000
5.6700 gtgt hvec1, 5, -3, 4, 0 hvec 1
5 -3 4 0
Arrays
gtgt coef1, 2 -4, 3 coef 1 2 -4
3
5
Basic Array Operations

• Addition/subtraction CAB where cij aijbij
• Multiplication/division CA . B where cij
  aijbij
• Exponentiation CA . 4 where cij aij4

gtgt CAB C -3 3 -7 9
B -4 1 -3 6
A 1 2 -4 3
gtgt CA.B C -4 2 12 18
gtgt CA./B C -0.2500 2.0000 1.3333
0.5000
gtgt CA.2 C 1 4 16 9
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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.
• when combining arrays, make sure they all have
  the same dimensions
• applies to vectors, 2D arrays, multi-dimensional
  arrays

gtgt A1 2 3 4 5 gtgt 2.A ans 2 4
6 8 10 gtgt 2A ans 2 4 6
8 10 gtgt B2 4 6 8 10 gtgt A.B ans 2
8 18 32 50 gtgt AB ??? Error using gt
Inner matrix dimensions must agree.
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More Notes on Array Operations

• Most Matlab functions will work equally well with
  both scalars and arrays (of any dimension)
• Use brackets to construct arrays
• Use colon notation (e.g., A(,2) or f(311) to
  index)

gtgt A1 2 3 4 5 gtgt sin(A) ans 0.8415
0.9093 0.1411 -0.7568 -0.9589 gtgt
sqrt(A) ans 1.0000 1.4142 1.7321
2.0000 2.2361
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Array Constructors

• Arrays are often read into Matlab from files or
  entered by the user
• But building arrays from scratch can be tedious
• Explicit
• Using Matlab array constructors

gtgt g(1)1 g(2)3 g(3)-4 g 1 3 -4
gtgt Aones(2,3) A 1 1 1 1
1 1 gtgt B-3ones(1,5) B -3 -3 -3
-3 -3 gtgt Czeros(2,3) C 0 0
0 0 0 0
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Lets Build Some Arrays
What will these produce?
gtgt A3eye(2,2) A 3 0 0 3 gtgt
Bdiag(1 2 3 4) B 1 0 0 0
0 2 0 0 0 0 3 0
0 0 0 4 gtgt Cdiag(1 2 1,1) C
0 1 0 0 0 0 2 0
0 0 0 1 0 0 0 0 gtgt
diag(A) ans 3 3
D magic(5) diag(D) diag(diag(D)) Z
magic(3),zeros(3,2), -ones(3,1) 4ones(2,4),
eye(2,2) Z(,3) mess 10rand(4,5) messy
10randn(4,5) test 1./(3ones(2,3)
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Vectors and Matrices

• Weve 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

Can you explain what, if anything, results from
these operations with vectors?
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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
• 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 Cramers Rule
• Other?
• Lets try a more abstract notation

OR
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A Simple Example-contd

• What do we mean by the for this form?
• 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

NOTE the number of columns in A must be equal
to the number of rows in B (N in this example)
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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 AB ?
  BA
• 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 3y6 and you need to find y
• The usual approach y6/32 (division by 3)
• Also useful y3-162 (multiplication by the
  inverse of 3)
• If we dont know how to divide, we can accomplish
  the same by using the notion of the inverse.
  Recall definition of inverse
• Turns out we know how to compute matrix inverses
  (but it requires a lot of computational effort)

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Lets Solve Our Problem Using Matlab
gtgt coef3 -2 1 4 coef 3 -2 1
4 gtgt inv(coef) Matlab has the inv()
function ans 0.2857 0.1429 -0.0714
0.2143 gtgt b14 -14' b 14 -14 gtgt
zinv(coef)b z 2 -4 gtgt coefz
Let's check our answer! ans 14 -14
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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

NOTEC\C1, and1anythinganything
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Lets Try This Out
coef 3 -2 1 4 gtgt b b 14
-14 gtgt zzcoef\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
• Now, using this, we take the transpose of our
  equation

where
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Lets Try It Out in Matlab
gtgt coefTcoef' coefT 3 1 -2
4 gtgt bTb' bT 14 -14 gtgt
zTbTinv(coefT) zT 2 -4 gtgt ALSO WE
CAN USE RIGHT DIVIDE gtgt zT2bT/coefT zT2
2.0000 -4.0000
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Other Matlab Matrix Functions

• So far weve only scratched the surface of
  Matlabs 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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Finally, What About Vectors?

• The matrix and array operations and functions can
  be used to manipulate vectors, but youll have to
  be careful
• Vector dot product
• ON YOUR OWN
• Vector magnitude?
• Vector cross product?

gtgt f1 2' f 1 2 gtgt g4 -3' g
4 -3 gtgt fdotgf'g fdotg -2
gtgt f1 2 f 1 2 gtgt g4 -3 g
4 -3 gtgt fdotgfg' fdotg -2 gtgt
gdotfgf' gdotf -2
Column vectors
Row vectors