KYM134 Computer Programming

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KYM134Computer Programming
Matrices and Operations

• Ankara University
• Chemical Engineering Department
• Assistant Prof. Dr. Refik SAMET

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Crash course on Matrices

• Definitions and Terms
• Special Matrices
• Matrix Algebra
• Determinant and Inverse
• Application Examples

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• Definitions and Terms

• Vector, matrix
• Element, subscript
• "Matrices of the same kind"

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Vector
A vector is a column of numbers consisting of n
numbers
The values shown by ai are the elements of
vector a.
A
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Matrix (Pl. Matrices)
A matrix is a rectangular array of numbers
consisting of m rows and n
columns.
The values shown by aij are the elements of
vector a.
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Matrix example
This matrix has 3 rows rows and 4 columns. We
say it is a 3 x 4 (3 by 4) matrix.
We denote the element on the second row and
fourth column with a2,4
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Subscripted expression
Elements are denoted using the indexed notation
ai,j

Subscripted expression
Column index
Row index
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"Matrices of the same kind"
Matrices A and B are of the same kind if and
only if A has as many rows as B, and
A has as many columns as B AMN BMN
are of the same kind.
We may often say that A and B are "of the same
order."
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• Special Matrices

• Row / column matrices
• Zero matrix
• Square matrix
• Diagonal matrix
• Unit matrix
• Symmetric matrix
• Transpose
• Orthogonal matrix

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Row matrix / column matrix
A matrix with one row is called a row matrix.
A matrix with one column is called a column
matrix.
Also called row-vector and column-vector.
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Zero Matrix
A zero matrix is a matrix with all elements 0.
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Square Matrix
A square matrix has the same number of rows and
columns.
The elements ai,i , with i 1,2,3,... are
called diagonal elements.
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Diagonal matrix
A diagonal matrix is a square matrix with all
non-diagonal elements 0.
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Unit matrix
A unit matrix is a diagonal matrix with all
diagonal elements 1.
Also called identity matrix - I.
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Symmetric matrix
A symmetric matrix is a square matrix where aI, J
aJ, I for all elements.
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Transpose

• The transpose of the m x n matrix A is shown by
  AT.

• AT is an n x m matrix,
• row i of A column i of AT for (i 1,2,3,
  .. n)

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Transpose (2)
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Transpose (3)
Matrix A' is the transpose of matrix A if and
only if The ith row of A the ith column of A'
for (i 1,2,3,..n)
Transposing a matrix means converting and m x n
matrix into an n x m matrix, by flipping the
rows and columns.
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Transpose (4)




4 x 3
3 x 4
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Transpose (5)



3 x 3
3 x 3
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Orthogonal matrix
An orthogonal matrix is a square matrix which
produces a unit matrix if it is multiplied by its
own transpose A ? AT I
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Properties of special matrices
A (AT)T (((AT)T )T )T
(A B)T (AT BT)
If A is a symmetric matrix, then A AT
If A is an orthogonal matrix, then A ? AT I
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• Matrix Algebra

• Scalar multiplication
• Matrix addition
• Matrix subtraction
• Vector addition
• Vector multiplication
• Matrix multiplication

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Scalar multiplication
To multiply a matrix by a number k, multiply each
element of the matrix by the number k.

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Matrix addition
To add two matrices, add elements of the first
to the corresponding element of the second.
!!! Addition is possible if matrices are of the
same order.
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Matrix subtraction
A - B To subract a matrix B from A, subtract
elements of B from the corresponding element of A.
!!! Subtraction is possible if matrices are of
the same order.
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Vector addition
To add two vectors, add the corresponding
elements.
The result is a vector of the same size.
!!! Addition is possible if vectors are of the
same size.
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Vector multiplication
You can multiply a row vector with a column
vector.
To multiply two vectors, multiply the
corresponding elements, then add the results.
The result is a single number.
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Vector multiplication (2)
Multiply the corresponding elements, then add the
results
2
?

2 (- 4) 0 2
4
0
!!! Multiplication is possible if vectors are of
the same order.
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Matrix multiplication
Let A be a k x m matrix and B be
an m x n matrix, Then the product A ? B C is
defined. C is a k x n matrix.
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Matrix multiplication
AKM ? BMN CKN

• The number of rows of C is equal to that of A,
  (which is K)
• The number of columns of C is equal to that of
  B (which is M).

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Matrix multiplication
!!! AB ? BA (Unless A, B are both NxN)
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Matrix multiplication (2)
Element ci j is the vector product of row i with
column j.

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Properties of matrix addition
Addition is associative ABC (AB) C
A (BC)
A 0 A 0 A (0 zero matrix)
Addition is symmetric A B B A
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Properties of Scalar multiplication
(k, s scalar numbers)
k (AB) kA kB
(ks) A kAsA
(ks) A k (sA)
(kA)T k AT
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Properties of Matrix multiplication
Multiplication is associative A?B?C (A?B) ?
C A ? (B?C)
Multiplication is distributive A?(B C) A?B
A?C
(A?B)T BT ? AT
A?0 0?A 0 (0 is the zero matrix)
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• Determinant and Inverse

• Minor
• Cofactor
• Determinant
• Inverse

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Inverse of a number

• Let n be a scalar (number).
• The inverse of the number is shown by n-1 1
  / n
• From algebra n n-1 1
• If n 0, then n-1 1 / 0
  (which is undefined)

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Inverse of a matrix

• Let A be a square matrix.
• The inverse of matrix A is shown by A-1
• By definition, AA-1 A-1A I,
  where I is the identity matrix.
• If ?A? 0, then the inverse A-1 is
  undefined.

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Determinant of a matrix
?A? is called the determinant of matrix
A. Sometimes written also as det(A) When
computed, ?A? turns into a single
number. We'll see how ?A? can be computed in a
short while.
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Finding the inverse of a matrix
Only square matrices have inverses.
To find the inverse of A , ... you need to find
?A?.
To find the determinant ?A?, ... you need to
find the cofactors of elements of A.
... you need to find the minors of ?A?
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Finding minor of element aij
The minor of element aij is the matrix obtained
by deleting row i and column j of A.

Minor(a12)
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Finding cofactor of element aij
The cofactor of element aij is Cofactor (aij)
Cij (-1)ij ?minor(aij)?
We need to find the cofactor matrix C computing
Cij for each aij . (This is a recursive process)
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Finding the inverse of A
The inverse of A A-1 1/?A? CT
Note that ?A? is a number ... so is 1 /
?A? .
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Finding the determinant of a 2x2 matrix
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Finding the cofactor matrix of a 2x2 matrix
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Finding the inverse of a 2x2 matrix
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Finding the determinant of a 3x3 matrix
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Properties of determinants
A matrix A and its transpose have the same
determinant ?A? ?AT?
The determinant of a diagonal matrix is the
product of the diagonal elements.
When we exchange two columns in A, or, when we
exchange two rows in A, A changes sign.
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• Matrix Algebra - Applications

• Linear equations
• Gaussian elimination

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Linear equations
A system of n equations, with n unknowns. a11
x1 a1n xn b1 a21 x1 a2n xn
b2 an1 x1 ann xn bn
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Linear equations (2)
Can be written as a matrix equation A X B
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Linear equations (3)
If the determinant A is nonzero, the equation
can be solved to produce n numerical values for x
that satisfy all the simultaneous equations

• Find A-1 (the inverse of A) (it must exist
  because A is nonzero)
• Multiply both sides of the equation by A-1
• A-1A X A-1 B
• Since A-1 A I
• X A-1 B

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Linear equations (3) Example 1
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Linear equations (3) Example 1
The equation becomes
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Solving linear equations Gaussian elimination
If we had a single equation, multiplying (or
dividing) both sides with the same number would
not change the equation 2x 12
1/2 (2x) 1/2 (12)
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Solving linear equations Gaussian elimination (2)
If we had a two equations, multiplying (or
dividing) both sides of one equation with the
same number would not change the equation 4x
2y 0 2x - y -4
Divide the first equation by 2, then add to the
second equation to eliminate y the resulting
system is 2x y 0 2x - y -
4 x -2 y 1
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Solving linear equations Gaussian elimination (3)
If we had a matrix equation, multiplying (or
dividing) both sides of a row with the same
number would not change the equation


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Solving linear equations Gaussian elimination (4)
Gaussian elimination is designed to solve a
general set of n equations having n unknowns.

• Two steps
• Forward Elimination - multiple of one equation is
  subtracted from another to eliminate unknowns
• Back Substitution last equation yields the
  unknown, substituting back into the other
  equations yields the rest.

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Solving linear equations Gaussian elimination (5)
Consider n equations in n unknowns a11 x1
a1n xn b1 a21 x1 a2n xn
b2 an1 x1 ann xn bn

• The solution vector X will not change the
  following row operations are performed
• Multiply or divide any equation (row) by a
  constant.
• Add or subtract any equation to another
  equation.

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Solving linear equations Gaussian elimination (6)
2x1x24x316 3x12x2x310 x13x23x316
Forward Elimination
2x1 x2 4x3 16 1/2 x2 - 5x3 -
14 x1 3x2 3x3 16
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Solving linear equations Gaussian elimination (7)
2x1 x2 4x3 16 1/2 x2 - 5x3 -14 x1
3x2 3x3 16
Forward Elimination
2x1 x2 4x3 16 1/2 x2 - 5x3 -
14 5/2 x2 x3 8
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Solving linear equations Gaussian elimination (8)
2x1 x2 4x3 16 1/2 x2 - 5x3 -14
5/2 x2 x3 8
Forward Elimination
2x1 x2 4x3 16 1/2 x2 - 5x3 -
14 26 x3 78
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Solving linear equations Gaussian elimination (7)
Back substitution
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MatricesandVectorsMatrix and Array Operations
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Matrices and Vectors

• Matrix
• MATLAB input command
• gtgt A 2 5 7 8 5 6 8 3 1 6 4 0

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Matrices and Vectors

• Matrix
• MATLAB input command
• gtgt B 2x log(x)sin(y) 5x 32i

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Special cases vectors and scalars

• A vector is a special case of a matrix, with just
  one row or one column
• gtgt u 1 3 9 produces a row vector
• gtgt v 1 3 9 produces a column vector
• A scalar does not need brackets.
• gtgt g 9.81
• Null Matrix
• gtgt X

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Continuation

• The following three commands are equivalent
• A 1 3 9 5 10 15 0 0 -5
• A 1 3 9
• 5 10 15
• 0 0 -5
• A 1 3 9 5 10 ...
• 15 0 0 -5
• ... usage is not limited to matrix input

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Indexing (or Subscripting)

• A 1 2 3 4 5 6 7 8 8
• A
• 1 2 3
• 4 5 6
• 7 8 8
• A(2, 3)
• ans
• 6
• A(3, 3) 9
• A
• 1 2 3

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Indexing (or Subscripting)

• B A(23, 13)
• B
• 4 5 6
• 7 8 9
• B A(23, )
• B
• 4 5 6
• 7 8 9
• B(, 2)
• B
• 4 6

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Dimensioning

• No explicit dimension declarations are required
• MATLAB creates a matrix just big enough.
• If the matrices B and C do not exist already
• B(2, 3) 5 produces

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Dimensioning

• C(3, 13) 1 2 3 produces

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Dimensioning

• C(3, 13) 1 2 3
• C
• 0 0 0
• 0 0 0
• 1 2 3
• size(C)
• ans
• 3 3
• m, n size(C)
• m
• 3

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Matrix Manipulation

• By specifying vectors as the row and column
  indices of a matrix one can reference and modify
  any submatrix.
• If A is a 10 X 10 matrix, B is a 5 X 10 matrix,
  and y is a 20 elements long row vector, then
• A(1 3 6 9, ) B(13, ) y(1 10)
• replaces 1st, 3rd, and 6th rows of A by the
  first 3 rows of B, and the 9th row of A by the
  first 10 elements of y.

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Matrix Manipulation

• Reshaping matrices
• As a vector All the elements of a matrix A can
  be strung into a single column vector b by the
  command b A() (A is stacked in vector B).
• As a differently sized matrix If A is m x n, it
  can be reshaped into a p x q matrix as long as
• m x n p x q
• Thus, for a 6 x 6 matrix A,
• gtgtreshape(A, 9, 4) transforms A into a 9 x 4
  matrix,
• gtgtreshape(A, 3, 12) transforms A into a 3 x 12
  matrix

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Matrix Manipulation

• Transpose of matrix A
• Obtained by typing A
• A2 3 6 7
• A
• 2 3
• 6 7
• B A'
• B
• 2 6
• 3 7

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Matrix Manipulation

• Transpose example
• u 0 1 2 3 4 5 6 7 8 9
• v u(36)'
• v
• 2
• 3
• 4
• 5

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Initialization

• Not necessary but advisable for
• Large matrices (To reserve a continuous block in
  memory for efficient processing).
• gtgt A zeros(m, n) initializes m x n matrix
• Dynamix matrices If the rows or columns of a
  matrix is computed in a loop you may initialize
  the matrix to a null mutrix (A ) and append
  rows or columns later

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Appending a row or column

• A 1 0 0 0 1 0 0 0 1
• A
• 1 0 0
• 0 1 0
• 0 0 1
• u 5 6 7
• u
• 5 6 7
• A A u
• A
• 1 0 0

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Appending a row or column

• A 1 0 0 0 1 0 0 0 1
• A
• 1 0 0
• 0 1 0
• 0 0 1
• v 2 3 4
• v
• 2
• 3
• 4
• A A v
• A

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Appending a row or column

• A 1 0 0 0 1 0 0 0 1
• A
• 1 0 0
• 0 1 0
• 0 0 1
• u 5 6 7
• u
• 5 6 7
• A A u'
• A
• 1 0 0 5

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Appending a row or column

• A 1 0 0 0 1 0 0 0 1
• A
• 1 0 0
• 0 1 0
• 0 0 1
• u 5 6 7
• u
• 5 6 7
• A A u
• ??? All matrices on a row in the bracketed
  expression must have the
• same number of rows.

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Appending a row or column

• B B B 1 2 3
• B
• 1 2 3
• B for k13, BB k k1 k2 end
• B
• B
• 1 2 3
• 2 3 4
• 3 4 5

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Deleting a row or column

• Any row(s) or column(s) of a matrix can be de-
  leted by setting the row or column to null vector
• A(2, ) deletes the 2nd row of matrix A
• A(, 35) deletes the 3rd through 5th
  columns
• A(1 3, ) deletes the 1st and the 3rd row
  of A
• U(5length(u)) deletes all elements of
  vector U except 1 through 4

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Utility matrices

• eye(m, n) returns an m x n matrix with
• 1s on the main diagonal
• zeros(m, n) returns an m x n matrix of zeros
• ones(m, n) returns an m x n matrix of ones
• rand(m, n) returns an m x n matrix of
• random numbers
• Above commands with a single argument, e.g.
  ones(m), produce square matrices of dimension m

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Utility matrices

• diag(v) generates a diagonal matrix with
• vector v on the diagonal
• diag(A) extracts the diagonal of matrix A
• as a vector
• diag(A, 1) extracts the first upper offdiagonal
• vector of matrix A
• A matrix can be built with many block matrices as
  well.

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Utility matrices

• eye(3)
• ans
• 1 0 0
• 0 1 0
• 0 0 1
• B ones(3) zeros(3, 2) zeros(2, 3) 4eye(2)
• B
• 1 1 1 0 0
• 1 1 1 0 0
• 1 1 1 0 0
• 0 0 0 4 0
• 0 0 0 0 4
• diag(B)'

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Utility Matrices

• d 2 4 6 8
• d1 -3 -3 -3
• d2 -1 -1
• D diag(d) diag(d1, 1) diag(d2, -2)
• D
• 2 -3 0 0
• 0 4 -3 0
• -1 0 6 -3
• 0 -1 0 8

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Creating Vectors

• v InitialValueIncrementFinalValue
• (If no increment is specified, increment
• is 1)
• a 010100 produces a 0 10 20 ... 100,
• u 210 produces a 2 3 4 ... 10
• No square brackets are required above.
• However, to force concatenation you need them
• u 110 33-219

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Creating Vectors

• linspace(a, b, n) generates a linearly spaced
  vector of length n from a to b.
• logspace(a, b, n) generates a logarithmically
  spaced vector of length n from 10a to 10b.
• u zeros(1, 1000) initializes a 1000 element
  long row vector
• v ones(10, 1) creates a 10 element long column
  vector of 1s.
• u linspace(0, 20, 5)
• u
• 0 5 10 15 20
• v logspace(0, 3, 4)
• v
• 1 10 100 1000

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Matrix and Array Operations

• AB or A-B is valid if A and B are of the
• same size
• AB is valid if As number of columns
• equals Bs number of rows
• A/B is valid, and equals A.B-1 for same
• size square matrices A B
• A2 makes sense only if A is square
• and equals AA

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Matrix and Array Operations

• In all the previous comands if B is replaced by a
  scalar, say w, the arithmetic operations are
  still carried out. Aw adds w to each element of
  A, wA (or Aw) multiplies each element of by w
  and so on.
• The right division This division is used to
  solve a matrix equation. In particular,
• X A\b solves the matrix equation A x b.

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Matrix and Array Operations

• Element-by-element multiplication, division and
  exponentiation between two matrices or vectors of
  the same size are done by preceding the
  corresponding arithmetic operators by a period
  (.)
• . element-by-element multiplication
• ./ element-by-element left-division
• .\ element-by-element right-division
• . element-by-element exponentiation
• . nonconjugated transpose

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Matrix and Array Operations

• A1 2 3 4 5 6 7 8 9
• x A(1, )'
• x
• 1
• 2
• 3
• x'x
• ans
• 14
• xx'
• ans

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Matrix and Array Operations

• v 1 2 3 4 5 6
• v
• 1 2 3 4 5 6
• 1./v
• ans
• 1.0000 0.5000 0.3333 0.2500
  0.2000 0.1667

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Matrix and Array Operations

• There is a big difference between A2 and A.2
• A 1 2 3 4 5 6 7 8 9
• A
• 1 2 3
• 4 5 6
• 7 8 9
• A2
• ans
• 30 36 42
• 66 81 96
• 102 126 150
• A.2

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Matrix and Array Operations

• Relational Operations
• lt less than
• lt less than or equal
• gt greater than
• gt greater than or equal
• equal
• not equal
• These operations result in a vector or matrix of
  the same size as the operands, with 1 where the
  relation is true and 0 where false.

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Matrix and Array Operations

• Relational operations
• x 1 5 3 7
• y 0 2 8 7
• k x lt y
• k
• 0 0 1 1
• k x gt y
• k
• 1 1 0 1

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Matrix and Array Operations

• Logical operations
• logical AND
• logical OR
• logical complement (NOT)
• xor exclusive OR

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Matrix and Array Operations

• x 0 5 3 7
• y 0 2 8 7
• m (xgty)(xgt4)
• m
• 0 1 0 0
• n xy
• n
• 0 1 1 1
• m (xy)
• m
• 1 0 0 0

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Matrix and Array Operations

• Built-in logical functions
• all true (1) if all elements of a vector are
  true
• any true (1) if any element of a vector is
  true
• exist true (1) if the argument (a variable or
• function) exists
• isempty true(1) for an empty matrix
• isinf true for all infinite elements of a matrix
• isfinite true for all finite elements of a matrix
• isnan true for all elements of a matrix that are
• Not-A-Number
• find finds indices of non-zero elements of a
  matrix

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Matrix and Array Operations

• x 7 -2 9 -1 -3 10
• all(xlt0)
• ans
• 0
• A 1 inf 2 3
• isfinite(A)
• ans
• 1 0
• 1 1
• x 0 2 5 7
• find(x)
• ans

104
Matrix and Array Operations

• Matrix functions
• expm(A) finds the exponential of matrix A, eA
• logm(A) finds log(A) such that A e log(A)
• sqrtm(A) finds A
• Array Counterparts that operate on each element
  of the input matrix
• exp
• log
• sqrt

105
Matrix and Array Operations

• A 1 2 3 4
• asqrt sqrt(A)
• asqrt
• 1.0000 1.4142
• 1.7321 2.0000
• Asqrt sqrtm(A) Thus AsqrtAsqrtA
• Asqrt
• 0.5537 0.4644i 0.8070- 0.2124i
• 1.2104- 0.3186i 1.7641 0.1458i
• exp_aij exp(A)
• exp_aij

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Character strings

• Character Strings
• gtgt message Leave me alone
• creates a vector of size 1 x 14
• To create a column vector of text objects, each
  text string must have exactly the same number of
  characters
• gtgt names John Ravi Mary Xiao
• Thus
• gtgt A Ali Veli
• generates an error

107
Matrix and Array Operations

• Character Strings
• char command pads the necessary spaces
• gtgt howdy char(Hi, Hello, Namaste)
• is valid.
• Character strings can be manipulated just like
  matrices.
• gtgt c howdy(2,) names(3,) produces
• Hello Mary as the output in variable c.

108
Matrix and Array Operations

• Charater Strings
• Check page 62 of youe book for a number of
  character functions.
• eval function evaluates text strings and
  execute them if they contain valid commands
• eval(x 5sin(pi/3)) is equivalent to
• x 5sin(pi/3)
• You make eval useful by using variables in the
  string.