TAILIEUCHUNG - Ebook Advanced engineering mathematics (2nd edition): Part 2

The book pedagogically develops a strong understanding of the mathematical principles and practices that today's engineers need to know. Its comprehensive instructional framework supports a conversational, down-to-earth narrative style, offering easy accessibility and frequent opportunities for application and reinforcement. | Chapter 11 The Eigenvalue Problem Introduction In this chapter we study the problem Ax Ax I where A is a given n X n matrix X is an unknown n X 1 vector and A is an unknown scalar. If we re-express 1 as Ax AIx where I is an n X n identity matrix then subtraction of AIx from both sides gives the equivalent equation A - AI x 0 2 which is a homogeneous system of n equations in the n unknown Xj where the coefficient matrix A - Al contains the parameter A. To be sure that 1 and 2 are clear let US write them out in scalar form for n 3 for example. Then 1 is the system 11211 12 2 2121 . a22x2 23213 A t 2 31 1 32212 33 3 x3- Subtracting the terms on the right from those on the left gives an - A x i 012212 13213 0 21 l 22 A C2 0 31211 32212 33 - A . t 3 0 which in matrix form is equation 2 . Of course we don t need to insert the I. We could re-express 1 correctly as Ax Ax 0. but it would not follow from the latter that A - A x 0 because subtraction of a scalar A from a matrix A is not defined. Hence the need to insert I. 541 542 Chapter 11. The Eigenvalue Problem From Chapter 10 we know that 2 is consistent because it necessarily admits the trivial solution X 0. However our interest in 2 shall be in the search for nontrivial solutions and we anticipate that whether or not nontrivial solutions exist will depend upon the value of A. Thus the problem of interest is as follows given the n X n matrix A find the value s of A if any such that 2 admits nontrivial solutions and find those nontrivial solutions. The latter is called the eigenvalue problem and is the focus of this chapter. The A s that lead to nontrivial solutions for X are called the eigenvalues or characteristic values and the corresponding nontrivial solutions for X are called the eigenvectors or characteristic vectors . The eigenvalue problem 1 or equivalently 2 occurs in a wide variety of applications such as vibration theory chemical kinetics stability of equilibria buckling of structures .

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