PHYS108 Matrices, Vectors,

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PHYS108 Matrices, Vectors, Scalars 09/09/09
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• Understand essentials of operator precedence.
• Be able to use square brackets, to create
  matrices.
• Be able to identify important built-in functions
  required to create and manipulate matrices.

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• To save command window input/output
• gtgt diary(lecture_09_09_09)

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• 2.1 Scalar Arithmetic Operations
•  Scalars
• add (), subtract(-), multiply (), divide (/),
  exponentiation ().
• BUT multiple operations 533? gt 153 or
  527 ?
• Precedence determines order of arithmetic
  operations. Rules are ordered from highest(1) to
  lowest precedence level (4)
•  
• Parentheses ().
• Exponentiation ().
• Multiplication () and division (/).
• Addition () and subtraction (-).
• Within each level, operators with equal
  precedence are evaluated in order from left to
  right.
• Such rules of precedence exist for ALL systems
  of computer arithmetic

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• gtgt 24/10 1st then
• ans 1.6000
• gtgt 5 49 1st then
• ans 41
• gtgt -9 -822 1st then then -
• ans -41
• gtgt 6 6/23 / then then
• ans 15
• gtgt 6 6/(23) 1st inside () then / then
  
• ans 7
• gt 224 2(24) 44 216
• ans 256 65536

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• Common Intrinsic MATH Functions
• Math Notation MATLAB name
• sinx sin(x)
• cosx cos(x)
• tanx tan(x)
• a abs(a)
• lnex log(x)
• log10x log10(x)
• vx sqrt(x)
• See
• fx gt MATLAB gt Mathematics gt Elementary
  Mathematics

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• 2.2 Creation of Matrices and Vectors
• Explicitly via square brackets,
• row vectors separate elements by spaces or
  commas
• gtgt v 3 2 1
• v 3 2 1
• column vectors separate elements by semicolons
• gtgt w 6 5 4
• w 6
• 5
• 4

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• Matrix gt rectangular table of numbers
• Ar,c gt r row index (start from 1)
  from top to bottom
• c column index (start from 1) from left
  to right

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• Separate rows by semicolons
• gtgt A 1 2 3 4 5 6 7 8 9 10 11 12
• A 1 2 3 4
• 5 6 7 8
• 9 10 11 12
• Create matrices from predefined vectors and
  matrices
• gtgt u vvv
• u 3 2 1
• 3 2 1
• 3 2 1

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• gtgt t vvv w
• t 3 2 1 6
• 3 2 1 5
• 3 2 1 4
• gtgt A t
• ans 1 2 3 4 3 2 1 6
• 5 6 7 8 3 2 1 5
• 9 10 11 12 3 2 1 4
• gtgt A t
• ans 1 2 3 4
• 5 6 7 8
• 9 10 11 12
• 3 2 1 6
• 3 2 1 5
• 3 2 1 4

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• Colon operator
• lower-value step upper-value
• step can be positive or negative
• gtgt x 00.251
• x 0 0.2500 0.5000 0.7500 1.0000
• gtgt y 1.0-0.250.0
• y 1.0000 0.7500 0.5000 0.2500 0

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• SOME Intrinsic Functions
• transpose(x) transpose of column vector is a
  row vector and transpose of a row vector is a
  column vector.
• gtgt transpose(w)
• ans 6 5 4
• gtgt transpose(v)
• ans 3
• 2
• 1
• Transpose of matrix switches row column indices
• Transpose(Aij) Aji

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• gtgt A 1 2 3 4 5 6
• A 1 2 3
• 4 5 6
• gtgt transpose(A)
• ans 1 4
• 2 5
• 3 6
• linspace(x1, x2, N) generates a row vectors with
  N equal spaced points between x1 and x2.

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• gtgt linspace(0.0, .01, 5)
  gt five-member row
• ans 0 0.2500 0.5000 0.7500 1.0000
• gtgt transpose(linspace(0.0, 1.0, 5)) gt
  five-member column
• zeros(m,n) gt m-by-n matrix of 0s.
• ones(n) gt n-by-n matrix of 1s.
• eye(n) gt n-by-n identity matrix.
• rand(n) gt n-by-n matrix of
  uniformly distributed random real numbers.
• randi(imax, m,n) gt returns an m-by-n matrix
  of random integers uniformly between 1 and
  imax.
• reshape(A, m, n) gt returns m-by-n matrix
  taken
• columnwise from A.

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• Additional functions to generate matrices
• magic(n)
• pascal(n)
• hilb(n)
• See gallery for 50 matrices with unusual
  properties
• fx gallery

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• gtgt zeros(3,2)
• ans 0 0
• 0 0
• 0 0
• gtgt ones(2,3)
• ans 1 1 1
• 1 1 1
• gtgt rand(3,4)
• ans
• 9.7540e-002 9.5751e-001 9.7059e-001
  8.0028e-001
• 2.7850e-001 9.6489e-001 9.5717e-001
  1.4189e-001
• 5.4688e-001 1.5761e-001 4.8538e-001
  4.2176e-001
• gtgt randi(1000, 4,3)
• ans 916 36 758
• 793 850 744
• 960 934 393

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gtgt x linspace(1, 24, 24) gtgt A reshape(x, 6,
4) A 1 7 13 19 2 8
14 20 3 9 15 21 4 10
16 22 5 11 17 23 6 12
18 24 gtgt A reshape(x, 4, 6) A 1 5
9 13 17 21 2 6 10 14
18 22 3 7 11 15 19 23
4 8 12 16 20 24 gtgt B
transpose(A) B 1 2 3 4 5
6 7 8 9 10 11 12 13
14 15 16 17 18 19 20 21
22 23 24
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• Functions to Manipulate Vectors Matrices
• Vector
• length(x) returns the number of elements of
  vector x.
• mean(x) returns the mean value of the
  elements of x.
• median(x) returns the median value of the
  elements of x.
• sort(x) sorts x in ascending order
• sort in descending order, sort(x,
  descend)
• sum(x) sum of the elements of x.
• min(x) returns the smallest element of x.
• Essential property of ALL such functions, applied
  to matrices, is that they treat each column of a
  matrix as a distinct vector.
•  

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• gtgt A 1 2 3 4 5 6 7 8 9
• gtgt sort(A, 'descend')
• ans 7 8 9
• 4 5 6
• 1 2 3
• gtgt sum(A)
• ans 12 15 18
• gtgt min(A)
• ans 1 2 3

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• If you want the relevant function applied to
  entire matrix, apply the function to A() -
• gtgt sum(A()) MATLAB TRICK
• ans 45
• gtgt max(A())
• ans 9
• gtgt sort(A(), 'descend')
• ans 9
• 8
• 7
• 6
• 5
• 4
• 3
• 2
• 1
• Sometimes one sees reference to A() where A is
  known to be a matrix. This operation transforms a
  two-dimensional matrix, A, into a corresponding
  one-dimensional vector, constructed in column
  major order
•  

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• Extract/Identify Elements via Subscripts and
  Colon Operators
• The relevant subscript/index is identified by an
  integer placed inside a pair of matched
  parentheses.
• gtgt y 026
• gtgt y(2)
• ans 2
• gtgt y(23)
• ans 2 4
• gtgt y(4 2 1)
• ans 6 2 0 

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• gtgt D 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
• D 1 2 3 4 5
• 6 7 8 9 10
• 11 12 13 14 15
• gtgt D(1,) identify 1st row
• ans 1 2 3 4 5
• gtgt D(, 5) identify 5th column
• ans 5
• 10
• 15
• gtgt D(23, 35)
• ans 8 9 10
• 13 14 15
• gtgt D(23, 35) -1 -2 -3 -4 -5 -6 Replace
  section
• D 1 2 3 4 5
• 6 7 -1 -2 -3
• 11 12 -4 -5 -6

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• Empty 0x0 matrix
•  
• The operator can be employed to remove
  columns/rows from matrices
•  
• gtgt C 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
  16
• gtgt C(2,) removes 2nd row
• C 1 2 3 4
• 9 10 11 12
• 13 14 15 16
• gtgt C(,1) removes 1st
  column
• C 2 3 4
• 10 11 12
• 14 15 16
•  
•  

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• Understand essentials of operator precedence gt
  () trumps everything else .
• Be able to use square brackets, , to create
  small vectors and matrices.
• Be able to identify some important built-in
  functions linspace, transpose, rand, randi,
  randn, min, max, size, sort, ones, zeros, eye
• Be familiar with colon operator.