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Matrix multiplication is particularly important When A Rn m and B Rm p , then their product C = AB Rn p is de ned as ci,j = m k=1 ai,k bk,j Note that the number of columns in A and the number of rows in B, referred to as the inner dimension of the product, must be identical; otherwise the two matrices cannot be multiplied Using the vectors and matrices de ned above, we have the following: AB A B = = 02396 63776 14300 13559 not a valid operation

FIGURE 3-7

Matrix multiplication does not commute: AB = BA Matrix division is unde ned Matrix multiplication does have the following useful formulas: (AB) C = (A + B) C = (ABC) = A (BC) AC + BC C B A Associative property Distributive property (B4) (B5) (B6)

In the certain derivations it will be convenient to insert appropriately dimensioned identity matrices The reader should be comfortable with formulas such as the following: ABC = AIm BIp C Ax = AIm x (B7) (B8)

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basic option chain, and you will nd this type of option quote provided by most online brokerage houses as well as most of the nancial media sites For more advanced trading, the basic option chain is missing vital information needed for effectively making option trading decisions If you are not currently using option analysis software for your trading, you might consider checking into the various offerings from software companies and Internet resources The Internet is an excellent resource for nding option pricing modeling applications You can nd offerings from many different sources including free software downloads, JavaScript applications, predesigned spreadsheets for Excel, and services offering fee-based software applications Many of these programs are relatively low or no cost, while others can be several hundred dollars a month with real-time data feeds The most important part of any software or online application is that you, as the trader or analyst, be able to perform an analysis and develop information over the basic option chain Throughout this book we demonstrate fee-based software-generated option chains, and models Figure 27 shows the expanded capability of software in presenting the relevant information You can see in Figure 27 Wal-Mart, Inc, option chain that we are able to expand our view by analyzing theoretical option value, implied volatility, and the Greeks (See below) By using a software analysis program you can more effectively lay out your potential strategy and gather more information regarding the risk and reward potential of the position You should also be aware of the limitations of these programs Remember that options models, including the ones we demonstrate throughout this book, are theoretical and based on assumed

assign a priority for fast file creation operations, described earlier, as part of the overall plan The Exadata Storage Server provides a rich set of metrics, which can give you a great deal of information about the operation of IORM at the category, database, or consumer group level For more details on these tracking options, please refer to the Exadata documentation

The subscript denoting the dimension will typically be dropped to simplify the notation, as it is usually clear from the context A square matrix A is an orthogonal matrix if and only if AA = A A is a diagonal matrix A square matrix A is an orthonormal matrix if and only if AA = A A = I Often authors use the term orthogonal to mean orthonormal Let Rb represent the vector transformation from reference a frame a to reference frame b The inverse transformation from reference frame b to reference frame a can be shown to be Ra = (Rb ) 1 = (Rb ) a a b Then (Rb ) Rb = Rb (Rb ) = I This shows that Rb and Ra are orthonora a a a a b mal matrix The scalar product of two vectors u, v RN is de ned to be

(B9)

.

The square of the Euclidean norm of a vector is de ned as the scalar product of a vector with itself: u 2 = u u The vector u is a unit vector if u = 1 A vector has both magnitude and direction, with magnitude de ned by u u and direction de ned by u The scalar product has the following properties: u v = v u=u v =v u u (v + w) = u v + u w u v = u v cos( ) (B10) (B11) (B12)

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