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where A - lI is sometimes called the characteristic matrix of A How does a determinant become a polynomial The best way to see this is with an example of a symmetric 2 2 matrix: 1 07 1 0 1 - l 07 A - lI = - l 0 1 = 07 1 - l 07 1 Now the determinant det(A - lI) = (1 - l)2 - 049, which is seen to be a quadratic (polynomial) in l Note that the highest power of l is 2, which is also the dimension of A For n-dimensional matrix A we would have an nth degree polynomial in l The eigenvalues are also called characteristic roots, proper values, latent roots, and so on These have fundamental importance in understanding the properties of a matrix.

ery complying with safety standards, and then encouraged its technical personnel to take part in schemes and courses organized by the industry associations, and site personnel to attend training courses. Such was the background to the accident.Work on a residential building was nearing completion, and the various subcontractors called in by The Harbour to do the nishing work were present on the site. Among them was The Light, a company specialized in the installation of electrical systems. In order to plan the schedule of work for the weeks leading up to the accident, meetings were held between the site foreman and the owner of The Light. Some days after the rst meeting, the owner of The Light came onto the site with two young apprentices to start work on the electrical installations. The work proceeded quite rapidly, and soon, after initially being present only occasionally the two apprentices came in almost every day. The owner of The Light, who liaised with the site foreman, increasingly absented himself in order to supervise work on another building site nearby, leaving the two apprentices on their own. The young electricians gained increasing familiarity with the site and began to work independently, even without instructions from the site foreman, and in relative isolation from other workers.

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When analyzing SEM traf c, you need to pay close attention to key dimensions. They are: Medium: For search, Medium should be either organic or cpc (for SEM traffic). Source: Sources are search engines referring traffic, such as google.com, yahoo, and bing. Campaign: Campaign relates to the unique campaign name you assigned in your tracking tags or inside AdWords. Keyword: Keyword is the search term that you bid for with the search engine (the actual search term used by the searcher rather than the bid term can be found by using a filter, as explained later in this chapter). Ad Content: This dimension should relate to the text ad you ve created (note that you have to use the utm_content tracking tag). User-Defined: This dimension is useful if you have applied the Actual Search Term Discovery filter covered later in this chapter. Geographic: The Country, Region, and City dimensions are useful for analyzing the performance of SEM providers, campaigns, ad groups, and keywords based on geography (for example, if you sell a coffee thermoses, you will probably see higher conversion rates from North Dakota than Florida during winter months). Other dimensions in custom reports: Other useful dimensions not listed in the standard cross-dimension menu or used in standard reports include Ad Slot, Ad Group, Ad Position, Visit Number, Depth of Visit, and Length of Visit. Likewise, there are a number of metrics that are critical to analyze when optimizing SEM efforts. They include:

For arbitrary square matrices, eigenvalues are complex roots of the characteristic polynomial After all, not all polynomials have real roots For example, the polynomial l2 + 1 = 0 has roots l1 = i and l2 = -i where i denotes -1 The eigenvalues are denoted by li where i = 1, 2, , n, are in nonincreasing order of absolute values For complex numbers, the absolute value is de ned as the square root of the product of the number and its complex conjugate The complex conjugate of i is -i, their product is 1, with square root also 1 Thus |i| = 1 and |-i| = 1, and the polynomial l2 + 1 = (l - l1)(l - l2) holds true As in the example above, the roots of a polynomial need not be distinct.

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