Matlab Chapter 2: Array and Matrix Operations

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MatlabChapter 2 Array and Matrix Operations
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What is a vector?

• In Matlab, it is a single row (horizontal) or
  column (vertical) of numbers or characters.
• Vector is one row OR one column
• Does not have to have anything to do with
  geometry
• Number of components is not restricted
• Transpose of row is column, of column is row

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Vectors in Matlab

• Various forms for entering row and column
  vectors

gtgt u1234 u 1 2 3 4
gtgt p1,2,3 p 1 2 3
gtgt v329 v 3 5 7 9
gtgt w47 w 4 5 6 7
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Transposing Vectors
gtgt u u 1 2 3 4 gtgt u' ans
1 2 3 4

• gtgt p
• p
• 1 2 3
• gtgt p'
• ans
• 1
• 2
• 3

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Augmenting Row Vectors

• gtgt p
• p
• 1 2 3
• gtgt v
• v
• 3 5 7 9
• gtgt zp v
• z
• 1 2 3 3 5 7 9

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Stacking Column Vectors
gtgt qut q 1 2 3 4
3 8 -4

• gtgt u
• u
• 1
• 2
• 3
• 4

gtgt t t 3 8 -4
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Functions to Generate Vectors

• linspace(a,b,c) produces c evenly spaced points
  between a and b
• gtgt linspace(2,4,5)
• ans
• 2.0000 2.5000 3.0000 3.5000
  4.0000
• logspace(a,b,n) computes n pts linearly between a
  and b, then uses them as exponents of 10
• gtgt logspace(1,3,4)
• ans
• 1.0e003
• 0.0100 0.0464 0.2154 1.0000

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Matrices

• A rectangular array of numbers or characters
• Rows numbered top to bottom
• Columns numbered left to right
• Size of array n x m
• number of rows is always stated first
• element of array x(k,j)
• row index is always stated firstkth row, jth col
• In general, indices can be negative or zero, but
  not in MATLAB
• Transpose of real matrix interchanges rows and
  columns

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Transpose Concatenate Matrices

• gtgt A1,2,34,5,6
• A
• 1 2 3
• 4 5 6
• A is transpose
• gtgt A'
• ans
• 1 4
• 2 5
• 3 6

• gtgt B4,58,9
• B
• 4 5
• 8 9
• A,B augments (concatenates rows)
• ans
• 1 2 3 4 5
• 4 5 6 8 9
• AB stacks (concatenates cols)
• ans
• 1 4
• 2 5
• 3 6
• 4 5
• 8 9

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Vector Addressing

• v 1,7,3,8,6,7,3
• v(4) returns the 4th elementgtgt v(4)
• ans 8
• Note parentheses, not square brackets!
• v(25) returns elements 2 through 5
• gtgt v(25)
• ans 7 3 8 6

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Vector Addressing cont.

• v 1,7,3,8,6,7,3
• v() returns the entire vector as a column
• gtgt v()
• ans
• 1
• 7
• 3
• 8
• 6
• 7
• 3

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Matrix Addressing

• A
• 3 6 8
• 1 5 2
• A(2,1) returns the element in the 2nd row, 1st
  column
• ans 1
• A(,3) returns the 3rd column
• ans 8 2
• A(2,) returns the 2nd row
• ans 1 5 2
• A(12,23) returns all elements in the 1st and
  2nd rows that are also in 2nd and 3rd columns

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Matrix Addressing

• A
• 3 6 8
• 1 5 2
• A(12,23) returns all elements in the 1st and
  2nd rows that are also in 2nd and 3rd
  columns ans 6 8
• 5 2
• Extract smaller array
• B A(12,1 3)B 3 8 1 2

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Arrays/Matrix Addressing

• Empty or null array
• A
• 3 6 8
• 1 5 2
• Remove a column
• A(,3) A 3 6 1 5
• Automatic enlargement
• A(3,5) 22A 3 6 0 0 0
• 1 5 0 0 0 0 0
  0 0 22

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Arrays/Matrix Addressing

• Negative increment reverses order
• p3,8,4,6
• rev p(end-11)rev 6 4 8 3
• A(n) returns the nth element of matrix A, going
  column by columnA 3 6 8
• 1 5 2
• A(4)ans 5

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Arrays/Matrix Addressing

• A() returns all elements as a column, going
  column by column A 3 6 8
  1 5 2
• A()ans 3 1 6 5
  8 2

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Automatic growth of a vector

• If vector already defined as row or column,
  assigning new element beyond current bounds ?
  automatically grows to fit with intervening
  elements set to 0
• x 12 x starts as 2-element column vector
• x(5) 24 x now has 5 elements. x(3) x(4)
  0x
• 1
• 2
• 0
• 0
• 24

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Automatic growth of a vector

• If variable is not yet vector, assigning element
  beyond 1 defaults to creating a row vector
• gtgtclear xx
• xx(4) 12 xx starts as a 4-element row vector
• xx
• 0 0 0 12

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Warning!

• Use clear to Avoid Errors
• Be careful
• suppose you simply want A to have u and v as
  columns
• A(,1) u
• A(,2) v
• A however, still has its previous columns!
• Use clear to avoid this
• clear A

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Vector and Matrix Functions
Play around with all these functions!
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Array Operations Same operation performed on
each corresponding element of array

• A
• 3 7 2
• 6 -1 5
• gtgt A3
• ans
• 6 10 5
• 9 2 8
• gtgt 2A
• ans
• 6 14 4
• 12 -2 10
• gtgt A.2
• ans
• 9 49 4
• 36 1 25

• B
• 4 6 1
• 3 9 2
• gtgt AB
• ans
• 7 13 3
• 9 8 7
• gtgt A.B
• ans
• 12 42 2
• 18 -9 10
• A./B
• ans
• 0.7500 1.1667 2.0000
• 2.0000 -0.1111 2.5000

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Math Functions Automatically Apply to Each Element
A 3 7 2 6 -1 5
gtgt sin(A) ans 0.1411 0.6570 0.9093
-0.2794 -0.8415 -0.9589
gtgt sqrt(A) ans 1.7321 2.6458
1.4142 2.4495
0 1.0000i 2.2361
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Vector Operations
v -4 5 2

• u
• 3 2 5

gtgt dot(u,v) ans 8 gtgt cross(u,v) ans
-21 -26 23
gtgt norm(u) ans 6.1644 gtgt this is the
Euclidean length or magnitude
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Matrix Product

• C AB is defined if and only if the number of
  columns in A equals the number of rows in B. Then
  C will have the same number of rows as A and the
  same number of columns as B
• (3x2)(2x4) produces a 3x4
• (3x2)(3x2) not defined
• (3x2)(2x3) produces a 3x3
• (2x3)(3x2) produces a 2x2

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Matrix Product CAB

• The (i,j) entry in C is the sum of the products
  of entries from the i-th row of A and the j-th
  column of B

A 4 5 -1 3 2 0
B 3 2 1 5 1 -1 0
3 2 5 1 4
gtgt AB ans 15 -2 3 31 11
4 3 21
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Matrix Power

• An means multiply A by itself n times.
• Can only be done if A is square
• A.n means raise each element of A to the n-th
  power
• Can be done with any array A

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Matrix Division

• If A and B are matrices, A/B is NOT a defined
  operation in linear algebra!
• Matlab uses both right (/) and left (\) operator
  symbols for special matrix operations we are not
  covering in this course!
• 1./A is an array operation creating a new array
  with each entry being the reciprocal of the entry
  in A
• A./B is an array operation creating a new array
  with each entry being the corresponding entry in
  A divided by the corresponding entry in B

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Special Matrices

• eye(n)nxn identity matrix
• eye(3) 1 0 0
• 0 1 0
• 0 0 1
• ones(m,n) mxn matrix with all entries 1
• zeros(m,n)mxn matrix with all entries 0