soln04 - 10.34 Fall 2006 Homework #4 - Solutions Problem 1:...

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10.34 – Fall 2006 Homework # 4 - Solutions Problem 1: Problem 3.A.3 in Beers’ textbook – Eigenvalue/vectors Part A: (see page 150-151 in Beers) We are asked is any of the matrices are known to have all real eigenvalues. Matrices A can be visually seen to have only real eigenvalues, this is because it is a real symmetric matrix. Matrix C can also be determined to have only real eigenvalues because: Trace(C) – 4*det(C) = 24 > 0 This requires a small calculation, but not the actual calculation of the eigenvalues. Part B: (see page 156-157 in Beers) The bounding of the eigenvalues involves the use of Gershgorin’s Theorem, which states that the eigenvalues of a matrix must lie within some range around the diagonal elements of the matrix. More specifically: λ i a k k = Γ k a j elements j k The summation is the sum of the modulus of all the off-diagonal elements for row k . Matrix A: Doing this analysis for matrix A, we find that the Gamma value for rows 1 – 4 are: 4, 5, 2, and 5, respectively. This results in the following bounds: 4 or 4 i 4 i i −≤ 25 or 3 i 7 6 i 7 i 33 or 0 i ≤ ≤ 6 i +≤ 15 or 6 i 4 Matrix C: Similarly for matrix C, we find Gamma values of: 2 and 1. 32 or 1 i i 5 2 i 5 11 or 2 i i −≤ ≤ 0 Note that in this case, the range between 0 and 1 is also excluded from the value eigenvalue space, but the problem asked for an upper and lower bound. Part C: For a unitary matrix, D -1 = D T . So if we can calculate the inverse and show it equals the transpose, then D is unitary. In order to determine the inverse, we can make an augmented matrix to solve D * D -1 = I . Similar to the idea of Gaussian elimination, we want to do row operations to turn D into I , and the matrix that was formerly I , will be the inverse of D . To accomplish this, we only need to do two things: exchange rows 1 and 2; then multiply row 2 by -1. The results are shown below. Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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0 1 0 100 0 0 1 1 0 0 100 Row operations 0 1 0 ⎯⎯⎯⎯⎯→ 0 1 0 001 0 0 1 0 10 1 0 0 01 By visual inspection, you can see that D -1 does equal D T .
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soln04 - 10.34 Fall 2006 Homework #4 - Solutions Problem 1:...

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