Question: For a matrix of order 20 whose eigenvalues are 1, 2, 3, ( ( ( ( 20, the characteristic equation can be written as (20m=1

For a matrix of order 20 whose eigenvalues are 1, 2, 3, ( ( ( ( 20, the characteristic equation can be written as (20m=1 (( - m) = 0, where the product notation is defined by (nk=1 bk ( b1b2( ( (bn. Use a spreadsheet or computer-algebra system to graph the characteristic polynomial for ( in the range ( = 0.9 to 20.1. Choose the scale on the vertical axis so that the points where the curve crosses the horizontal axis are clearly visible. Now add the quantity 1 ( 10-8(19 to the characteristic polynomial, graph the altered characteristic polynomial, and notice what has happened to the roots of the characteristic polynomial. If you see fewer than 20 roots for the altered polynomial, explain where the missing roots are. In the original characteristic polynomial, the coefficient of (19 is the sum of the integers 1 to 20, which is 210, so the change made was less than one part in 1010 in the (19 coefficient, yet it caused a major change in the eigenvalues.

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