MATLAB Basics

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MATLAB Basics

• With a brief review of linear algebra
• by Lanyi Xu
• modified by D.G.E. Robertson

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1. Introduction to vectors and matrices

• MATLAB MATrix LABoratory
• What is a Vector?
• What is a Matrix?
• Vector and Matrix in Matlab

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What is a vector

• A vector is an array of elements, arranged in
  column, e.g.,

X is a n-dimensional column vector.
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• In physical world, a vector is normally
  3-dimensional in 3-D space or 2-dimensional in a
  plane (2-D space), e.g.,

, or
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• If a vector has only one dimension, it becomes a
  scalar, e.g.,

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Vector addition

• Addition of two vectors is defined by

Vector subtraction is defined in a similar
manner. In both vector addition and subtraction,
x and y must have the same dimensions.
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Scalar multiplication

• A vector may be multiplied by a scalar, k,
  yielding

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Vector transpose

• The transpose of a vector is defined, such that,
  if x is the column vector

its transpose is the row vector
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Inner product of vectors

• The quantity xTy is referred as the inner product
  or dot product of x and y and yields a scalar
  value (or x y).

If xTy 0 x and y are said to be orthogonal.
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• In addition, xTx , the squared length of the
  vector x , is
• The length or norm of vector x is denoted by

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Outer product of vectors

• The quantity of xyT is referred as the outer
  product and yields the matrix

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• Similarly, we can form the matrix xxT as

where xxT is called the scatter matrix of vector
x.
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Matrix operations

• A matrix is an m by n rectangular array of
  elements in m rows and n columns, and normally
  designated by a capital letter. The matrix A,
  consisting of m rows and n columns, is denoted as

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Where aij is the element in the ith row and jth
column, for i1,2,?,m and j1,2,,n. If m2 and
n3, A is a 2?3 matrix
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• Note that vector may be thought of as a special
  case of matrix
• a column vector may be thought of as a matrix of
  m rows and 1 column
• a rows vector may be thought of as a matrix of 1
  row and n columns
• A scalar may be thought of as a matrix of 1 row
  and 1 column.

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Matrix addition

• Matrix addition is defined only when the two
  matrices to be added are of identical dimensions,
  i.e., that have the same number of rows and
  columns.

e.g.,
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• For m3 and nn

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Scalar multiplication

• The matrix A may be multiplied by a scalar k.
  Such multiplication is denoted by kA where

i.e., when a scalar multiplies a matrix, it
multiplies each of the elements of the matrix,
e.g.,
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• For 3?2 matrix A,

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Matrix multiplication

• The product of two matrices, AB, read A times B,
  in that order, is defined by the matrix

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The product AB is defined only when A and B are
comfortable, that is, the number of columns is
equal to the number of rows in B. Where A is m?p
and B is p?n, the product matrix cij has m rows
and n columns, i.e.,
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For example, if A is a 2?3 matrix and B is a 3?2
matrix, then AB yields a 2?2 matrix, i.e.,
In general,
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For example, if
and
, then
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and
Obviously,
.
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Vector-matrix Product

• If a vector x and a matrix A are conformable, the
  product yAx is defined such that

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For example, if A is as before and x is as follow,
, then
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Transpose of a matrix

• The transpose of a matrix is obtained by
  interchanging its rows and columns, e.g., if

Or, in general, Aaij, ATaji.
then
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Thus, an m?n matrix has an n?m transpose. For
matrices A and B, of appropriate dimension, it
can be shown that
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Inverse of a matrix

• In considering the inverse of a matrix, we must
  restrict our discussion to square matrices. If A
  is a square matrix, its inverse is denoted by A-1
  such that

where I is an identity matrix.
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An identity matrix is a square matrix with 1
located in each position of the main diagonal of
the matrix and 0s elsewhere, i.e.,
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It can be shown that
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MATLAB basic operations

• MATLAB is based on matrix/vector mathematics
• Entering matrices
• Enter an explicit list of elements
• Load matrices from external data files
• Generate matrices using built-in functions
• Create vectors with the colon () operator

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gtgt x1 2 3 4 5 gtgt A 16 3 2 13 5 10 11 8 9
6 7 12 4 15 14 1 A 16 3 2 13
5 10 11 8 9 6 7 12
4 15 14 1 gtgt
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Generate matrices using built-in functions

• Functions such as zeros(), ones(), eye(),
  magic(), etc.

gtgt Azeros(3) A 0 0 0 0 0
0 0 0 0 gtgt Bones(3,2) B 1
1 1 1 1 1
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gtgt Ieye(4) (i.e., identity matrix) I 1 0
0 0 0 1 0 0 0 0 1
0 0 0 0 1 gtgt Amagic(4) (i.e.,
magic square) A 16 2 3 13 5
11 10 8 9 7 6 12 4 14
15 1 gtgt
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Generate Vectors with Colon () Operator
The colon operator uses the following rules to
create regularly spaced vectors jk is the
same as j,j1,...,k jk is empty if j gt
k jik is the same as j,ji,j2i, ...,k jik
is empty if i gt 0 and j gt k or if i lt 0 and j lt
k where i, j, and k are all scalars.
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Examples
gtgt c05 c 0 1 2 3 4
5 gtgt b00.21 b 0 0.2000 0.4000
0.6000 0.8000 1.0000 gtgt d8-13 d 8
7 6 5 4 3 gtgt e82 e
Empty matrix 1-by-0
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Basic Permutation of Matrix in MATLAB

• sum, transpose, and diag
• Summation
• We can use sum() function.
• Examples,

gtgt Xones(1,5) X 1 1 1 1
1 gtgt sum(X) ans 5 gtgt
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gtgt Amagic(4) A 16 2 3 13 5
11 10 8 9 7 6 12 4
14 15 1 gtgt sum(A) ans 34 34
34 34 gtgt
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Transpose
gtgt Amagic(4) A 16 2 3 13
5 11 10 8 9 7 6 12
4 14 15 1 gtgt A' ans 16 5
9 4 2 11 7 14 3 10
6 15 13 8 12 1 gtgt
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Expressions of MATLAB

• Operators
• Functions

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Operators
Addition-Subtraction Multiplication / Division
\ Left division Power ' Complex conjugate
transpose ( ) Specify evaluation order
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Functions
MATLAB provides a large number of standard
elementary mathematical functions, including
abs, sqrt, exp, and sin.
Useful constants
pi 3.14159265... i Imaginary unit (
) j Same as i
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gtgt rho(1sqrt(5))/2 rho 1.6180 gtgt
aabs(34i) a 5 gtgt
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Basic Plotting Functions plot( )
The plot function has different forms, depending
on the input arguments. If y is a vector,
plot(y) produces a piecewise linear graph of
the elements of y versus the index of the
elements of y. If you specify two vectors as
arguments, plot(x,y) produces a graph of y versus
x.
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Example, x 0pi/1002pi y sin(x) plot(x,y)
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Multiple Data Sets in One Graph
x 0pi/1002pi y sin(x) y2
sin(x-.25) y3 sin(x-.5) plot(x,y,x,y2,x,y3)
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Distance between a Line and a Point

• given line defined by points a and b find the
  perpendicular distance (d) to point c
• d
• norm(cross((b-a),(c-a)))/norm(b-a)