PROOF. We can always find one nonzero eigenvectorfor each eigenvalueofBy the preceding theorem, the setis linearly independent. Thusisdiagonalizable by the Diagonalization theorem.

2305. THE EIGENVALUE PROBLEMCaution:Just because thematrixhas fewer thandistinct eigenvalues, youmay not conclude that it is not diagonalizable.An example that illustrates this caution is part (a) of Example 5.1.13.5.2 Exercises1. Given each matrixbelow, find a matrixsuch thatis diagonal. Use thisto deduce a formula for(a)(b)(c)2. Determine if the following matrices are diagonalizable with a minimum of calcula-tion.(a)(b)(c)3. For each of the following matricesfind the characteristic polynomialandevaluate(This means that the matrixreplaces every occurrence ofand theconstant termis replaced by)(a)(b)(c)4. Suppose thatis an invertible matrix which is diagonalized by the matrixthat is,is a diagonal matrix. Use this information to find a diagonalization for5. Adapt the proof of Theorem 5.2.11 to prove that if eigenvectorsaresuch that for any eigenvalueof, the subset of all these vectors belonging toislinearly independent, then the vectorsare linearly independent.6. Suppose that the kill rateof Example 5.2.2 is viewed as a variable positive param-eter. There is a value of the numberfor which the eigenvalues of the correspondingmatrix are equal.(a) Find this value ofand the corresponding eigenvalues by examining the character-istic polynomial of the matrix.(b) Use the available MAS (or CAS) to determine experimentally the long term behaviorof populations for the value offound in (a).Your choices of initial states shouldinclude7. The thirteenth century mathematician Leonardo Fibonacci discovered the sequenceof integerscalled theFibonacci sequence. These numbers have a wayof turning up in many applications. They can be specified by the formulas

5.2. SIMILARITY AND DIAGONALIZATION231(a) Letand show that these equations are equivalent to the matrixequationsandwhere(b) Use part (a) and the diagonalization theorem to find an explicit formula for thethFibonacci number.8.Calculate second, third and fourth powers of the matrixBased on these calculations, make a conjecture about the form ofwhereisany positive integer.9. Show that any upper triangular matrix with constant diagonal is diagonalizable ifand only if it is already diagonal.Hint:What diagonal matrix would such a matrix besimilar to?10. Letbe atransition matrix of a Markov chain whereis not the identitymatrix.(a) Show thatcan be written in the formfor suitable realnumbers(b) Show thatandare eigenvectors for(c) Use (b) to diagonalize the matrixand obtain a formula for the powers of11. Show that ifis diagonalizable, then so is12. Letsquare matrices of the same size withinvertible.(a) Show that(b) Show that(c) Use (a) and (b) to show that ifis a polynomialfunction, then13. Prove the Cayley-Hamilton theorem for diagonalizable matrices; that is, show thatifis the characteristic polynomial of the diagonalizable matrixthensatisfiesits characteristic equation, that is,Hint:You may find Exercise 12 andCorollary 5.2.4 very helpful.

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