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where is a complex nth root of unity, and n is a power of 2 Notice how simple this matrix is to describe: its (j, k)th entry (starting row- and column-count at zero) is jk Multiplication by M = Mn ( ) maps the kth coordinate axis (the vector with all zeros except for a 1 at position k) onto the kth column of M Now here s the crucial observation, which we ll prove shortly: the columns of M are orthogonal (at right angles) to each other Therefore they can be thought of as the axes of an alternative coordinate system, which is often called 68

Think of the number line again, where the integers are points above and below a number reflector plane that goes through 0 As you move upward, the numerical value increases; as you go downward, it decreases As you get farther in either direction from the number reflector, the absolute value increases

There is one indicator that is more powerful than all the others Without a doubt, this indicator has the final word That indicator is the market itself, represented by any of the major indexes, including the Dow Jones Industrial Average, Nasdaq, S&P 500, Russell 2000, or Wilshire 5000, to name the most popular The best way to show you is by looking at the chart below of the S&P 500 (SPX) with a three-year time period

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To multiply, move out When you add an integer to itself, you multiply it by 2 When you add an integer to itself twice, you multiply it by 3 When you add an integer to itself 3 times, you multiply it by 4 You can keep going with this Here s an example of what happens when the integer is positive:

Figure 28 The FFT takes points in the standard coordinate system, whose axes are shown here as x1 , x2 , x3 , and rotates them into the Fourier basis, whose axes are the columns of Mn ( ), shown here as f1 , f2 , f3 For instance, points in direction x 1 get mapped into direction f1

5+5=5 2 5+5+5=5 3 5+5+5+5=5 4 Here s what happens when the integer is negative: 5 + ( 5) = 5 2 5 + ( 5) + ( 5) = 5 3 5 + ( 5) + ( 5) + ( 5) = 5 4

the Fourier basis The effect of multiplying a vector by M is to rotate it from the standard basis, with the usual set of axes, into the Fourier basis, which is de ned by the columns of M (Figure 28) The FFT is thus a change of basis, a rigid rotation The inverse of M is the opposite rotation, from the Fourier basis back into the standard basis When we write out the orthogonality condition precisely, we will be able to read off this inverse transformation with ease: Inversion formula Mn ( ) 1 =

$SPX (Daily) 109531 1500 1400 1300 1200 109531 1000 900 800 700

12 8 Finish here 4 Start here Multiply by 3 Add to itself 2 times Get 3 times as far from 0 Start here Multiply by 5 Add to itself 4 times Get 5 times as far from 0 4 8 Finish here 12

Figure 5-1 On top, 2 is multiplied by 3 On the bottom,

1 1 n Mn ( )

Suppose p is an integer and n is a positive integer From the above facts, you can see that whenever you multiply p by n, you add p to itself (n 1) times On the number line, you move away from the number reflector (n 1) times by a distance equal to the absolute value of p If your starting integer p is above the number reflector, you move up; if your starting integer p is below the number reflector, you move down Figure 5-1 illustrates how this works for 2 3 and 2 5 The number reflector is shown as a horizontal, dashed line What if n is negative instead of positive To multiply p by n in this situation, first take the additive inverse (negative) of your starting integer p, and then move away from the number reflector |n | 1 times by a distance equal to the absolute value of p Figure 5-2 shows how this works for 2 ( 3) and 2 ( 5) When you multiply two quantities, you get a product In a multiplication problem, the first quantity (the one to be multiplied) is sometimes called the multiplicand, and the second quantity (the one you are multiplying by) is sometimes called the multiplier More often, they are both called factors

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