Matrices

1
Matrices

• Data StructuresECE/CS 268
• Andrew Rupp

2
Outline

• Definition
• Matrix Manipulations
• Binary Matrices
• Sparse Matrices/Vectors
• Noninteger Keys
• Examples
• References

3
Definition

• A two dimensional array of elements
• Array of numbers or a table of variables
• An array of mathematical elements subject to
  certain algebraic laws and used notably to
  express the conditions in which a system of
  linear equations has a solution.

4
Matrix Manipulations

• Gauss-Jordan Elimination
• Row Reduced Echelon Form
• Multiplication
• Transpose
• Determinant
• Inverse

5
Gauss-Jordan
6
Row Reduced Echelon Form

• If a row has nonzero entries, then the first
  nonzero entry is 1, called the leading 1 in this
  row.
• If a column contains a leading 1, then all the
  other entries in that column are zero.
• If a row contains a leading 1, then each row
  above contains a leading 1 further to the left.

I
7
Multiplication

• The number of columns in the 1st Matrix must be
  equal to the number of rows in the 2nd Matrix
• Pouring rows of A down the columns of B

If A is m x p and B is p x n, then the product AB
must be m x n.
8
Multiplication
9
Determinant
10
Transpose

• Put the values from the columns into rows

11
Inverse

• A-1adj A / det A

12
Properties

• Noncommutative
• ABBA
• There are exceptions AIIAA
• Associative
• (AB)CA(BC)
• Distributive -
• A(CD)ACAD
• (AB)CACBC

13
Matrices in Java

• Defining A Matrix
• double d new double34
• String word new String93
• double price
• 1.25,1.09,0.58,0.25
• 1.35,0.98,0.62,0.35
• 1.40,1.12,0.43,0.19

14
Matrices in Java

• We think matrices are stored in memory the way we
  write a matrix on paper however the matrix is
  stored in a line.

15
Binary/Bit Matrix

• A binary/bit matrix is a matrix that holds 0/1,
  true/false, or yes/no.
• This uses less memory than a regular matrix.

16
Sparse Matrices/Vectors

• Matrix or a Vector that most values are null or
  0.
• The values that are not 0 are the only values
  stored.

17
Noninteger Keys

• Can store a map in matrix form using the
  MapMatrix constructor
• MapMatrix()
• MapMatrix(Object d)
• Methods
• Get(Object key1,Object key 2)-
• returns an Object
• set(Object key1, Object key2, Object2)

18
Examples

• Memory Game from Programming 2 with Bitterman
• Cards cnew Cards66

19
Examples

• Rain Game
• The Game of Life

20
Summary

• Matrices occur in a number of mathematical
  problems.
• Java provides a primitive matrix facility like
  the array data type.
• Sparse Matrix is a way to conserve space.
• A matrix can be a map of maps like the MapMatrix
  constructor.
• Bit Matrix is a way to store 1/0s using a lot
  less memory.

21
References

• Signals and Systems Continuous and Discrete
• Elementary Differential Equations Fourth Edition
• Linear Algebra with Applications
• Classic Data Structures in Java
• New Webster Dictionary and Thesaurus
• Wade Apels Memory Game