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single most powerful and useful feature of all the drawn objects, so it is crucial to understand their purpose. The handles of any drawn object are the places where the endpoints were established. Once you draw a trendline, only the line is shown, and no special mark is given to the endpoints. However, once you select Move from the popup menu (or once you double-click on the trendline, which is actually a faster way to achieve the same thing), the two handles will appear as small squares (Figure 7.11). When you point your mouse to a handle, the mouse pointer changes from an arrow to a hand, indicating it is ready for you to move the handle. You can drag the handle anywhere in the chart that you like. One of the reasons this is so useful is that you can create very precisely laid trendlines, which is important to good charting (Figure 7.12). You might want to make sure that the Snap to OHLC values during drawing checkbox is checked in the preferences menu (see Figure 3.18 back in 3) to make precise placement of handles easier. If you point at the trendline but do not click any button (this is known as a mouseover ), then information about the trendline will appear as a tool tip (Figure 7.13). This tool tip will identify the object (in this case,

To prove this, take matrix elements of the left side:

(ii'I{A X B){C X D)IiJ') = L(ii'IA X Blkk')(kk'IC X DIJJ')

= L(iIAlk)(kIClJ) L(i'IBlk')(k'IClJ')

(iIAClJ)(i'IBDlj') (ii'l (AC)

(BD )IJJ')

A generalization of (15.22) can be proved in the same way:

(A X B X ... X C){D X Ex .. XF) = (AD) X (BE) X ... X (CF) (15.:::

We now introduce some special matrices in terms of which VI' V2 , and V3 may t'C conveniently expressed. Let the three familiar 2 X 2 Pauli spin matrices t'C denoted by X, Y, and Z:

FIGURE 7.11 Double-clicking on a trendline gives you access to the two handles, which allow for careful placement of the line.

For subsequent saves, you can click the Save button ( ) on the Quick Access toolbar to quickly save the file.

X 2 = 1,

y 2 = 1,

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XY + YX = 0,

YZ + ZY = 0,

(15.25

XY= iZ, YZ = iX,

Let three sets of 2 X 2 matrices X a , Va' Za (a = 1, ... , n) be defined afollows*:

FIGURE 7.14 By clicking on the small triangle in the trendline tool you can access the features of the tool.

(n factors) (n factors) (n factors)

{15.26

Note: Another way to save a file is to press + . If this is the first time the file has been saved, Office launches the Save As dialog box.

[Xa,X p] = [Va' Yp ] = [Za,Zp] = [X a, Yp ] = [X a, Zp] = [Va' Zp] =

{15.27 1

For any given a the 2 n X 2 n matrices X a , Va' Za formally satisfy all the (15.25).

*The matrices Xa , Y , Za are familiar in quantum mechanics. For example, for a system of ' a nonrelativistic electrons the spin matrices for the ath electron are precisely Xa , Ya , and Za'

The following identity holds for any matrix X whose square is the unit matrix e 8X = cosh () + X sinh () (15.28) where () is a number. The proof is as follows. Since xn = X if n is odd, e 8X =

n=O n!

Trendline ) as well as the starting and ending values of the endpoints in terms of both price and date. There is usually a variety of ways to get to the same function, in order to save time. The ability to alter a trendline s style, color, or extension is available through right-clicking the line as well as clicking the trendline tool itself (Figure 7.14). There are two ways to access the popup menu using the trendline tool: You can either hold down the mouse button on the tool for a few moments, or you can click the small triangle in the tool s lower-right corner to get instant access. You can also access many of the functions of the toolbar through the Drawing Tools menu itself (Figure 7.15). The choices on this menu provide the functions shown in Table 7.1.

L - xn = L -

n even n!

n odd n!

= cosh () + X sinh ()

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