Matlab Training Session 2: Matrix Operations and Relational Operators

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Matlab Training Session 2Matrix Operations and
Relational Operators
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• Course Outline
• Weeks
• Introduction to Matlab and its Interface (Oct. 23
  2006)
• Fundamentals (Operators)
• Fundamentals (Flow)
• Importing Data
• Functions and M-Files
• Plotting (2D and 3D)
• Statistical Tools in Matlab
• Analysis and Data Structures
• Course Website
• http//www.queensu.ca/neurosci/Matlab Training
  Sessions.htm

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• Week 2 Lecture Outline
• Fundamentals of Matlab 1
• Week 1 Review
• B. Matrix Operations
• The empty matrix
• Creating multi-dimensional matrices
• Manipulating Matrices
• Matrix Operations
• C. Operators
• Relational Operators

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Week 1 Review
Working with Matrices c 5.66 or c 5.66
c is a scalar or a 1 x 1 matrix  x
3.5, 33.22, 24.5 x is a row vector or a
1 x 3 matrix  x1 2 5 3
-1 x1 is
column vector or a 4 x 1 matrix A 1 2 4
2 -2 2 0 3 5 5 4 9
A is a 4 x 3 matrix
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Week 1 Review

• Indexing Matrices
• A m x n matrix is defined by the number of m rows
  and number of n columns
• An individual element of a matrix can be
  specified with the notation A(i,j) or Ai,j for
  the generalized element, or by A(4,1)5 for a
  specific element.
• Example
• gtgt A 1 2 4 56 3 8 2 A is a
  4 x 2 matrix
• gtgt A(1,2)
• Ans 6

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Week 1 Review

• Indexing Matrices
• Specific elements of any matrix can be
  overwritten using the matrix index
• Example
• A 1 2 4 5
• 6 3 8 2
• gtgt A(1,2) 9
• Ans
• A 1 2 4 5
• 9 3 8 2

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Week 1 Review

• Indexing Matrices
• A 1 2 4 5
• 6 3 8 2
• The colon operator can be used to index a range
  of elements
• gtgt A(13,2)
• Ans 1 2 4

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Matrix Indexing Cont..

• Indexing Matrices
• A 1 2 4 5
• 6 3 8 2
• The colon operator can index all rows or columns
  without setting an explicit range
• gtgt A(,3)
• Ans 4 8
• gtgt A(2,)
• Ans 6 3 8 2

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B. Matrix Operations
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Matrix Operations

• Indexing Matrices
• An empty or null matrix can be created using
  square brackets
• gtgt A
• TIP The size and length functions can quickly
  return the number of elements and dimensions of a
  matrix variable

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Matrix Operations

• Indexing Matrices
• A 1 2 4 5
• 6 3 8 2
• The colon operator can can be used to remove
  entire rows or columns
• gtgt A(,3)
• A 1 2 5
• 6 3 2
• gtgt A(2,)
• A 1 2 5

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N Dimensional Matrices

• A 1 2 4 5 B 5 3 7 9
• 6 3 8 2 1 9 9 8
• Multidimensional matrices can be created by
  concatenating 2-D matrices together
• The cat function concatenates matrices of
  compatible dimensions together
• Usage cat(dimensions, Matrix1, Matrix2)

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N Dimensional Matrices

• Examples
• A 1 2 4 5 B 5 3 7 9
• 6 3 8 2 1 9 9 8
• gtgt C cat(3,1,2,4,56,3,8,2,5,3,7,91,9,9,8)
• gtgt C cat(3,A,B)

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Matrix Operations

• Scalar Operations
• Scalar (single value) calculations can be can
  performed on matrices and arrays
• Basic Calculation Operators
• Addition
• - Subtraction
• Multiplication
• / Division
• Exponentiation

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Matrix Operations

• Scalar Operations
• Scalar (single value) calculations can be
  performed on matrices and arrays
• A 1 2 4 5 B 1 C 5
• 6 3 8 2 7
• 3
• 3
• Try
• A 10
• A 5
• B / 2
• AC

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Matrix Operations

• Scalar Operations
• Scalar (single value) calculations can be
  performed on matrices and arrays
• A 1 2 4 5 B 1 C 5
• 6 3 8 2 7
• 3
• 3
• Try
• A 10
• A 5
• B / 2
• AC What is happening here?

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Matrix Operations

• Matrix Operations
• Matrix to matrix calculations can be performed on
  matrices and arrays
• Addition and Subtraction
• Matrix dimensions must be the same or the
  added/subtracted value must be scalar
• A 1 2 4 5 B 1 C 5 D 2 4
  6 8
• 6 3 8 2 7
  1 3 5 7
• 3
• 3
• Try
• gtgtA B gtgtA C gtgtA D

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Matrix Operations

• Matrix Multiplication
• Built in matrix multiplication in Matlab is
  either
• Algebraic dot product
• Element by element multiplication

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Matrix Operations

• The Dot Product
• The dot product for two matrices A and B is
  defined whenever the number of columns of A are
  equal to the number of rows of b
• A(x1,y1) B(x2,y2)

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Matrix Operations

• The Dot Product
• The dot product for two matrices A and B is
  defined whenever the number of columns of A are
  equal to the number of rows of b
• A(x1,y1) B(x2,y2)

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Matrix Operations

• The Dot Product
• The dot product for two matrices A and B is
  defined whenever the number of columns of A are
  equal to the number of rows of b
• A(x1,y1) B(x2,y2)

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Matrix Operations

• The Dot Product
• The dot product for two matrices A and B is
  defined whenever the number of columns of A are
  equal to the number of rows of b
• A(x1,y1) B(x2,y2) C(x1,y2)

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Matrix Operations

• The Dot Product
• A(x1,y1) B(x2,y2) C(x1,y2)
• A 1 2 B 1 D 2 2 E 2 4
  3 6
• 6 3 7 2 2
• 3
• 3
  
• Try
• gtgtA D
• gtgtB E
• gtgtA B

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Matrix Operations

• Element by Element Multiplication
• Element by element multiplication of matrices is
  performed with the . operator
• Matrices must have identical dimensions
• A 1 2 B 1 D 2 2 E 2
  4 3 6
• 6 3 7 2 2
  
• 3
• 3
• gtgtA . D
• Ans 2 4
• 12 6

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Matrix Operations

• Matrix Division
• Built in matrix division in Matlab is either
• Left or right matrix division
• Element by element division

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Matrix Operations

• Left and Right Division
• Left and Right division utilizes the / and \
  operators
• Left (\) division
• X A\B is a solution to AX B
• Right (/) division
• X B/A is a solution to XA B
• Left division requires A and B have the same
  number of rows
• Right division requires A and B have the same
  number of columns

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Matrix Operations

• Element by Element Division
• Element by element division of matrices is
  performed with the ./ operator
• Matrices must have identical dimensions
• A 1 2 4 5 B 1 D 2 2 2 2 E
  2 4 3 6
• 6 3 8 2 7 2
  2 2 2
• 3
• 3
• gtgtA ./ D
• Ans 0.5000 1.0000 2.0000 2.5000
• 3.0000 1.5000 4.0000 1.0000

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Matrix Operations

• Element by Element Division
• Any division by zero will be returned as a NAN in
  matlab (not a number)
• Any subsequent operation with a NAN value will
  return NAN

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Matrix Operations

• Matrix Exponents
• Built in matrix Exponentiation in Matlab is
  either
• A series of Algebraic dot products
• Element by element exponentiation
• Examples
• A2 A A (Matrix must be square)
• A.2 A . A

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Matrix Operations

• Shortcut Transposing Matrices
• The transpose of a matrix is the matrix formed by
  interchanging the rows and columns of a given
  matrix
• A 1 2 4 5 B 1
• 6 3 8 2 7
• 3
• 3
• gtgt transpose(A) gtgt
  B
• A 1 6
  B 1 7 3 3
• 2 3
• 4 8
• 5 2

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Matrix Operations
Other handy built in matrix functions
Include inv() Matrix inverse det()
Matrix determinant poly() Characteristic
Polynomial kron() Kronecker tensor product
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C. Relational Operators
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Relational Operators

• Relational operators are used to compare two
  scaler values or matrices of equal dimensions
• Relational Operators
• lt less than
• lt less than or equal to
• gt Greater than
• gt Greater than or equal to
• equal
• not equal

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Relational Operators

• Comparison occurs between pairs of corresponding
  elements
• A 1 or 0 is returned for each comparison
  indicating TRUE or FALSE
• Matrix dimensions must be equal!
• gtgt 5 5
• Ans 1
• gtgt 20 gt 15
• Ans 1

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Relational Operators
A 1 2 4 5 B 7 C 2 2 2 2
6 3 8 2 2 2 2
2 Try gtgtA gt B gtgt A lt C

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Relational Operators

• The Find Function
• The find function is extremely helpful with
  relational operators for finding all matrix
  elements that fit some criteria
• A 1 2 4 5 B 7 C 2 2 2 2
  D 0 2 0 5 0 2
• 6 3 8 2 2
  2 2 2
• The positions of all elements that fit the given
  criteria are returned
• gtgt find(D gt 0)
• The resultant positions can be used as indexes to
  change these elements
• gtgt D(find(Dgt0)) 10 D 10 2
  10 5 10 2

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Relational Operators

• The Find Function
• A 1 2 4 5 B 7 C 2 2 2 2
  D 0 2 0 5 0 2
• 6 3 8 2 2
  2 2 2
• The find function can also return the row and
  column indexes of of matching elements by
  specifying row and column arguments
• gtgt x,y find(A 5)
• The matching elements will be indexed by (x1,y1),
  (x2,y2),
• gtgt A(x,y) 10
• A 1 2 4 10
• 6 3 8 2

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Getting Help

• Help and Documentation
• Digital
• Accessible Help from the Matlab Start Menu
• Updated online help from the Matlab Mathworks
  website
• http//www.mathworks.com/access/helpdesk/help/tech
  doc/matlab.html
• Matlab command prompt function lookup
• Built in Demos
• Websites
• Hard Copy
• Books, Guides, Reference
• The Student Edition of Matlab pub. Mathworks Inc.