Matrices

1
Matrices
2
What is a Matrix

• An arrangement of numbers in rows and columns
• A mathematical structure

3
Using Matrices

• A coat company has 4 factories for manufacturing
  coats and sends them to 5 stores. The cost of
  sending one coat from a factory to a store is
  given in this table

• The amount of coats that each factory can
  manufacture for each store is given in this table

• What is total cost of transporting from a factory
  to a store ?

4
Matrix Operators in MATLAB

• Transpose Turns rows into columns and vice versa
• Colon used to obtain subparts of the matrix
• What is the total cost of transportation from
  factory 3
• size() gives the size of the matrix
• Replacing values in matrices
• Into ones ones()?
• Into zeros zeros()?
• Into values from other matrices, vectors or
  scalars
• Duplicating rows/columns
• Deleting rows and columns

5
Some Special Matrices

• Identity Matrix
• Diagonal Matrices
• Tridiagonal matrices

6
Matrix Manipulation

• diag extracts or creates diagonal matrices
• fliplr flips from left to right
• flipud flips from top to bottom
• rot90 rotates
• tril extracts the lower triangular part
• eye creates identity matrix

7
MATLAB Functions for Matrices

• sum
• mean
• max
• min

8
Vectorization of nested fors

• Multiple for loops can be vectorized by matrix
  operations
• Example If a regular fixed payment P is made n
  times a year to repay a loan of amount A over a
  period of k years, where the annual interest rate
  is r, then P is given by
• PrA(1r/n)(nk)/(n(1r/n)(nk)-1)?

9
Multidimensional Arrays

• Matrices can have more than two dimension
• Think of 3D arrays as a matrix in each page
• Used in color (RGB) pictures

10
Matrix---Numerical Entities

• Matrix Multiplication
• CAB
• C(ij)Sum(a(ik)b(kj)) k1m
• Matrix Multiplication generally is not
  commutative
• AAI (I is the identity matrix)?
• Some Other Functions
• inv---Inverse of matrix (sort of like dividing by
  the matrix)?
• lu---decomposing the matrix into upper and lower
  triangular parts
• svd--decomposing the matrix, to save storage
  space

11
Solving Equations

• 2x3y 4z9
• xy-z1
• 2x-y-z0
• What values of x,y,z satisfy all the three
  equations?
• We can use inv() to solve

12
From Numbers to Letters

• Strings are just arrays of letters
• Use the same vector operations as numbers
• For arithmetic operations letters are considered
  as numbers
• fprintf can be used for string manipulation
• Fun things to do
• Detect palindromes
• Substitute letters in strings
• Concatenate strings
• How many three lettered words can you make with
  a,o,c,t,r ?

13
Other String Functions

• blanks
• deblanks
• int2str
• num2str
• ischar
• lower
• Upper
• 2D strings

14
Eval Function

• eval() evaluates a MATLAB function
• Can be used to for interactive plotting
• lasterr() returns string containing last error
  message

15
Summary

• Sections covered
• 6.1 (except 6.1.14), 6.2,6.3 6.4, 6.5,6.6
  (except 6.6.2)?
• What we learned
• transpose operator, deleting columns, ones,
  zeros,size(), entering values, repmat, sum,
  min,mx, mean, all,any,diag,fliplr, flipud,
  rot90,tril, eye, matrix multiplication, svd, lu,
  inv,blanks,deblanks,lower, upper,ischar, int2str,
  num2str