Organizing Information with Matrices

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Organizing Information with Matrices

• Chapter 6.3 Solving Problems with Matrices,
  Graphs and Diagrams
• Mathematics of Data Management (Nelson)
• MDM 4U
• Authors Gary Greer and James Gauthier
• (with K. Myers)

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Definition

• according to our textbook, a matrix is a
  rectangular array of numbers set out in rows and
  columns.
• other definitions include

• biological matrix

• geological matrix

• fictional (?) virtual reality network
• 1970s British sci-fi television (e.g., The
  Deadly Assassin)
• William Gibsons novels (e.g., Neuromancer)
• recent films by the Wachowski Brothers

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The Dimensions of a Matrix

• In general, if a matrix A has m rows and n
  columns, it is called an m-by-n matrix.
• The variables m and n represent the dimensions of
  the matrix.
• In other words, the dimensions of a matrix
  correspond to the number of rows and columns in
  the matrix.
• The entry, a, in row i and column j is
  represented as aij

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Example

• A is a 3 x 3 matrix (i.e., m 3 and n 3)

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General Example

• A is an m x n matrix
• Note that the location scheme is like the
  Battleship Game

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Row and Column Operations

• The row sum of a matrix is the sum of the
  elements in a single row of a matrix.
• Similarly, the column sum of a matrix is the sum
  of the elements in a single column of a matrix.
• See pp. 348 349 of the text for an example that
  calculates the row sum and the column sum of a
  matrix.

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Example of row and column sums

• The sum of the first row is 11
• The sum of the second row is 31
• The sum of the first column is 12
• The sum of the second column is 20

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Multiplication of a Matrix by a Scalar

• In physics, a scalar is a quantity with magnitude
  but no direction.
• In terms of Chapter 6, we can think of a scalar
  as an individual number outside a matrix.
• When a matrix is multiplied by a scalar, the
  operations are simple
• each entry in the matrix is multiplied by the
  scalar.

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Example Scalar-Matrix Multiplication
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Addition and Subtraction of Matrices

• Addition and subtraction of matrices is only
  defined when the matrices involved are of equal
  dimension.
• Two matrices, A and B, may be added together by
  adding the corresponding entries in each matrix.

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Addition...
12
Subtraction...
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General form

• If matrix A and matrix B have the same
  dimensions, then...
• The sum S A B is the matrix formed by adding
  the entries of matrix A to the corresponding
  entries of matrix B, where
• sij aij bij
• The difference matrix D A B is formed by
  subtracting the entries of matrix B from the
  corresponding entries of matrix A, where
• dij aij bij

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Exercises / Homework

• Homework
• page 353, 1, 2
• page 354, 6
• page 355, 8a
• Also... examine the web links in this
  presentation

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Matrix Multiplication

• Chapter 6.4 Solving Problems with Matrices,
  Graphs and Diagrams
• Mathematics of Data Management (Nelson)
• MDM 4U
• Authors Gary Greer and James Gauthier
• (with K. Myers)

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Product of Two Matrices C AB

• The product, C, of matrix A and matrix B is
  defined only if A is an (m x k) matrix and B is a
  (k x n) matrix.
• In this case, C will be an (m x n) matrix.
• This means in order for C to be defined, the
  number of columns of A has to be equal to the
  number of rows of B.
• So, we may multiply A2x3 and B3x4 to produce C2x4
  We cannot multiply A2x3 and B4x3

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Example Row times Column

• Lets start by multiplying a row matrix and a
  column matrix with the same number of entries.
• m 1 and n 1 (i.e., A1xk times Bkx1)
• The result will be a single number (which, if one
  insists that the product of two matrices always
  must be a matrix, may be viewed as a 1x1 matrix).
• The element of the i-th row and the j-th column
  of the product is found by multiplying the i-th
  row of A by the j-th column of B.

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Example (continued)
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Now... A2x4 times B4x3
2102444
3122748
182140

5244980
10305688
15366396
70
80
90

158
184
210
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Matrix Multiplication (more)

• Given two matrices A and B, where A is an (m x k)
  matrix and B in a (k x n) matrix...
• The inner product is the number that results from
  the sum of the products of each row entry from
  matrix A with its corresponding column entry from
  matrix B.
• Therefore, if C AB, each entry in matrix C is
  the inner product of a row from matrix A and the
  corresponding column from matrix B.

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Transition Matrices

• A transition matrix is used to represent the
  number of edges that connect the vertices of a
  directed graph.
• Transition matrices may be used to represent
  paths through a maze, predator-prey relationships
  or any other interrelationship that can be
  represented by a digraph.
• If T is a transition matrix for a digraph, then
  Tn shows all paths of length n between the
  vertices of the graph.

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Example Building A Transition Matrix
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Exercises / Homework

• Read the examples
• pages 360 368
• Exercises
• page 369, 1, 2, 4
• page 370, 8, 9, 10
• page 372, 14