SolSec 4.2 - Problems and Solutions for Section 4.2 (4.17...

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Problems and Solutions for Section 4.2 (4.17 through 4.30) 4.17 Calculate the square root of the matrix M = 13 10 10 8 Hint : Let calculate and compare to M ab bc MM 12 2 // ;. = () Solution: Given: M = 13 10 10 8 If M / = , then MMM a b ab bc ab bc b c == = +− −− + = 22 13 10 10 8 This yields the 3 nonlinear algebraic equations: ab bc 13 10 8 += There are several possible solutions but only one that makes M positive definite which is a = 3, b = c = 2 as determined below in Mathcad. Choosing these values results in M 32 / =
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4.18 Normalize the vectors 1 2 0 5 01 ,, . . first with respect to unity (i.e., 1 = x x T ) and then again with respect to the matrix M (i.e., 1 = Mx x T ), where M = 30 1 2 . . Solution: (a) Normalize the vectors x 1 1 1 2 11 5 = == α xx T Normalized: x 1 1 5 1 2 0 4472 0 8944 = = . . x 2 2 0 5 5 = T Normalized: x 2 0 1 = x 3 3 002 = . . . T Normalized: x 3 50 1 2 1 1 0 7071 0 7071 = = = . . . . (b) Mass normalize the vectors
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x xx 1 1 1 2 11 11 4 = == α T M . Mass normalized: x 1 1 11 4 1 2 0 2962 5923 = = . . . x 2 0 5 = 2 50 T M x 2 1 50 0 5 1 2 0 1 0 0 7071 = = = . x 3 3 01 0 052 = . . . xM x T Mass normalized: x 3 1 0 052 0 4385 0 4385 = = . . . . .
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4.19 For the example illustrated in Figure P4.1 with 0 3 2 1 = = = c c c , calculate the matrix ˜ K ? Solution: From Figure 4.29, m m x kk k k x 1 2 12 2 22 3 0 0 0 + +− −+ = ˙˙ ˜ ˜ // / / / / KM K M m m k k m m K mk k m m k mm == = + () −− 1 2 2 3 1 2 1 1 1 2 2 1 2 1 0 0 0 0 2 21 1 23 km k k + Since K K K T ~ , ~ ~ = is symmetric. Using the numbers given in problem 4.2 yields ˜ K = 31 16 This is obviously symmetric.
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4.20 Repeat Example 4.2.5 using eight decimal places. Does P T P = 1, and does PKP T ˜ == [] Λ diag 1 2 ωω 2 2 exactly?
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SolSec 4.2 - Problems and Solutions for Section 4.2 (4.17...

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