Chapter 3 - Vectors & Matrices

Vectors & Matrices

• Scalar: a quantity that can be expressed as a single number

  • Ex. Height of a person

  • Ex. 3, -17, $\pi$ ,$\sqrt{-1}$ , 2 + 3 i, ….

• Vector: a sequence of numbers, symbols or functions

  • Ex. Dimensions of a box, coordinates of a point

  • MATLAB Ex. v = [3] , v = [1 2 3] , v = [x y z] , v = [x, x²,x³]

• Vectors in MATLAB have dimensions

  • Ex. v = [3] → 1 × 1 → 1 row x w column

  • Ex. v = [1 2 3] → 1 × 3 → 1 row x 3 columns

• Vectors can be either row vectors or column vectors.

Creating Vectors in MATLAB

• Vectors can be created in MATLAB in several ways:

  • Directly generate a vector:

    • Row Vector: v = [1 2 3] or v = [1, 2, 3] (, separates row elements)

    • Column Vector: v = [1; 2; 3] (; separates row elements)

  • Convert a row vector to a column vector:

    • v = v’ (Transpose function)

  • Using a sequence:

    • v = [1 : 3] = [1 2 3] or v = [1 : 2 : 5] = [1 3 5] (: will be referred to as the “colon operator”)

    • Use when you know the initial and final values and the increment

      • (start) : (end)    (interval of 1)

      • (start) : (interval) : (end)

  • Using linspace:

    • v = linspace(1, 10, 3) = [1.00 5.50 10.00]

    • Use when you know start and end, and number of elements in the array.

        linspace(start, end)
        linspace(start, end, n) 

Indexing Vectors in MATLAB

• Array indexing (MATLAB uses 1-based indexing)

  • Once we have created an array in MATLAB, we can read from or write to a single element at a time.

  • To do this, we reference the element’s position in the array including indicies.

v = [0 2 4 8]
v(1) = 0
v(2) = 2
v(3) = 4
v(4) = 8

• Reading information from Vectors

  • Access the third element of a vector v: x = v (3) → x = 4

• Writing information to vectors

  • Modify the third element of vector v: v(3) = 6 → v = [0 2 6 8]

Some Common Vector Operations

• Magnitude

x = norm(v)   

• Length

x = length(v)

• Sum the elements of a vector

x = sum(v)

• Minimum element of the vector

[value, index] = min(v)

• Maximum element of the vector

[value, index] = max(v)

Matrices

• Matrix: a rectangular array of number, symbols or functions

• A vector i essentially a 1-dimentional matrix → 1 row or 1 column

• A matric is characterized by its number of rows and columns

  • Row: horizontal set of elements

  • Column: vertical set of elements

    

• Special (Banded) Matrices

  

Creating Matrices in MATLAB

• Directly generate a matrix:

M = [1, 2, 3; 4, 5, 6; 7, 8, 9]

• From a collection of row vectors:

M = [r1; r2; r3]

• From a collection of column vectors:

M = [c1 c2 c3]

• Using a sequence:

M = [1 : 3; 4 : 6; 7 : 9]

Creating Special Matrices in MATLAB