Philosophiae Naturalis Principia Matlabematica

Damian Gordon

Getting to Matlab

Magic Matrix

MAGIC Magic square.

MAGIC(N) is an N-by-N matrix constructed from the integers

1 through N^2 with equal row, column, and diagonal sums.

Produces valid magic squares for N = 1,3,4,5,...

Identity Function

>> eye (4)

ans =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

Upper Triangle Matrix

» a = ones(5)

a =

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

» triu(a)

ans =

1 1 1 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 0 1

Lower Triangle Matrix

» a = ones(5)

a =

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

» tril(a)

ans =

1 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 0 1 1 1 1 1

Hilbert Matrix» hilb(4)

ans =

1.0000 0.5000 0.3333 0.2500

0.5000 0.3333 0.2500 0.2000

0.3333 0.2500 0.2000 0.1667

0.2500 0.2000 0.1667 0.1429

Inverse Hilbert Matrix» invhilb(4)

ans =

16 -120 240 -140

-120 1200 -2700 1680

240 -2700 6480 -4200

-140 1680 -4200 2800

Toeplitz matrix.

TOEPLITZ

TOEPLITZ(C,R) is a non-symmetric Toeplitz matrix having C as its

first column and R as its first row.

TOEPLITZ(R) is a symmetric (or Hermitian) Toeplitz matrix.

-> See also HANKEL

Summary of Functions

• magic• eye(4)• triu(4)• tril(4)• hilb(4)• invhilb(4) • toeplitz(4)

- magic matrix

- identity matrix

- upper triangle

- lower triangle

- hilbert matrix

- Inverse Hilbert matrix

- non-symmetric Toeplitz matrix

Dot Operator

• A = magic(4); b=ones(4);

• A * B

• A.*B

• the dot operator performs element-by-element operations, for “*”, “\” and “/”

Concatenation

• To create a large matrix from a group of smaller ones

• try– A = magic(3)– B = [ A, zeros(3,2) ; zeros(2,3), eye(2)]– C = [A A+32 ; A+48 A+16]– Try some of your own !!

Subscripts

• Row i and Column j of matrix A is denoted by A(i,j)

• A = Magic(4)

• try– A(1,4) + A(2,4) + A(3,4) + A(4,4)

• try– A(4,5)

The Colon Operator (1)

• This is one MatLab’s most important operators

• 1:10 means the vector– 1 2 3 4 5 6 7 8 9 10

• 100:-7:50– 100 93 86 79 72 65 58 51

• 0:pi/4:pi– 0 0.7854 1.5708 2.3562 3.1416

The Colon Operator (2)

• The first K elements in the jth column is– A(1:K, j)

• Sum(A(1:4, 4)) is the sum of the 4th column

or

• Sum(A(:, 4)) means the same