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Problem 1. Find the eigenvalue,eigenvector,geometric and algebraic multiplicity of eigenvalue, of each of the following linear transformation. And determine whether it could be diagonalizable or not. If diagonalizable, find eigenbasis. (1) V 3 dimensional. With basis e1 e2 e3 , and T : V −→ V such that T e1 e2 e3 = e1 e2 e3 1 1 1 1 1 1 1 1 1 (2) V 3 dimensional. With basis e1 e2 e3 , and T : V −→ V such that T e1 e2 e3 = e1 e2 e3 1 0 1 1 1 1 1 0 1 (3) V 4 dimensional. With basis e1 e2 e3 e4 , and T : V −→ V such that T e1 e2 e3 e4 = e1 e2 e3 e4 1 2 0 0 3 6 0 0 2 4 5 1 5 7 4 2 (3) V 4 dimensional. With basis e1 e2 e3 e4 , and T : V −→ V such that T e1 e2 e3 e4 = e1 e2 e3 e4 0 0 0 0 1 0 0 0 2 4 1 0 5 7 1 (4) V 4 dimensional. With basis e1 e2 e3 e4 , and T : V −→ V such that T e1 e2 e3 e4 = e1 e2 e3 e4 0 0 0 0 2 1 0 0 5 1 2 0 5 Problem 2. Find the eigenvalue and eigenvector for the following matrices A. Determine the algebraic multiplicity and geometric multiplicity. Determine whether it could be diagonalizable, if it could, please find P such that P −1AP is diagonal matrix. (1) A = 1 1 1 0 1 0 0 1 0 (2) A = 4 0 −2 0 1 0 1 0 1 (3) A = −9 4 4 −8 3 4 −16 8 7 1 3 3 1 1 Problem 3. For each of matrix A in the following excercise, find the matrix P, such that P −1AP is an upper triangular matrix. You can leave P as a product of two matrices. I can do an example for you. (0) M = 0 −1 −2 4 3 2 −1 0 2 We found 0 −2 1 is an eigenvector for 2. Thus we construct other vector 1 0 0 and 0 1 0 , you see these 1 will make sure the position of them covers 0 in your eigenvector, and lets calculate 0 −1 −2 4 3 2 −1 0 2 0 −2 1 = 0 −2 1 × 2 0 −1 −2 4 3 2 −1 0 2 1 0 0 = 0 4 −1 = 0 −2 1 × −1 + 0 1 0 × 2 0 −1 −2 4 3 2 −1 0 2 0 1 0 = −1 3 0 = 1 0 0 × −1 + 0 1 0 × 3 Thus, let P = 0 1 0 1 0 1 2 0 0 , we have MP = P 2 −1 −1 2 3 So P −1MP = 2 −1 0 −1 2 3 You see, P −1MP is block upper triangular. Now we made more accurate. We treat the right lower block. M2 = 0 −1 2 3 , We found it has eigenvalue 1 and has an eigen vector  −1 1 , Now choose this basis P2 = −1 1 1 0 We calculate M2 −1 1 = −1 1 M2 1 0 = 0 2 = 1 0 × 2 + −1 1 × 2 So M2P2 = P2 1 2 0 2 Thus −1 1 1 0 −1 0 −1 2 3 −1 1 1 0 = 1 2 2 So 1 −1 1 1 0 −1 2 −1 0 −1 2 3 1 −1 1 1 0 = 2 ∗ ∗ 1 2 2 And remember that 2 −1 0 −1 2 3 = 0 1 0 1 0 1 2 0 0 −1 0 −1 −2 4 3 2 −1 0 2 0 1 0 1 0 1 2 0 0 So Let Q = 0 1 0 1 0 1 2 0 0 1 −1 1 1 0 Then Q−1MQ would be an upper triangular matrix. With understanding of the above example, do the excercises in the next pages. (1) 6 19 −43 1 3 − 5 1 3 − 6 (2) 10 −3 28 −23 −6 60 − 7 −2 19 (3) 12 − 9 − 2 20 15 2 3 2 2 Problem 4. Suppose V is 10-dimensional linear space over C, T : V −→ V is a linear transformation has characteristic polynomial (λ − 1)3 (λ − 2)5 (λ + 2)(λ + 1)n . And dim(Im(T − I)) = 9 , dim(Im(T − 2I)) = 8 , dim(Im(T − 2I) 2 ) = 6 Suppose (v1) is a basis for Ker(T − I), and v2 v3 is a basis for Ker(T − 2I). v4 is an eigenvector for -2, and v5 is an eigenvector for -1/ 8 (0) What is n? Why? (1) Find all eigenvalue and eigenvector, geometric multiplicity, algebraic multiplicity for T 2 (2) What is the minimal polynomial for T? (3) Express T −1 as a polynomial of T. (4) Plot power-rank curve for linear transformation T 4 − 5T 2 + 4I Problem 5. Find all diagonalizable square matrix A, such that all its eigenvalue are the same. Problem 6. Suppose A is an 2 × 2 matrix, and tr(A) = 0, det(A) = −1, Calculate A20150628 Problem 7. Suppose A is an 2 × 2 matrix, and tr(A) = 3, tr(A2 ) = 5 Calculate det(A) Problem 8. Suppose A is an n × n matrix, and An = 0, Calculate tr(A) and det(A Problem 9. Suppose A =   7 19 −44 8 32 −70 4 15 −33 And each of the following matrices are all Commute with A. B1 = −2 −17 37 −4 −16 38 −2 −9 21 B2 = −7 −23 58 −16 −54 134 −8 −27 67 B3 = −2 −12 27 −4 −15 36 −2 −8 19 B4 = 3 7 −17 4 15 −34 2 7 −16 B5 = 1 5 −10 0 2 −2 0 1 −1 B6 = 12 38 −88 16 62 −140 8 30 −68 (1) Find the eigenvalue and eigenvector of A (2) Using the fact that they are commute with A. Calculate with your tricks on piece of draft paper. Then write eigenvalues of Bi and corresponding eigenvectors here. (3) Does these B commute each other? Why?(Because A have different eigenvalues.) Problem 10. Suppose A is an m × n matrix and B is an n × m matrix. Show that the matrix P = AB and Q = BA have the same non-zero eigenvalue. And for each non-zero eigenvalue λ 6= 0, the algebraic multiplicity of λ in P are the same with in Q, and so is geometric multiplicity. (This statement only true for non-zero eigenvalues. Hint: remember rank(I-AB)=rank(I-BA) and det(I-AB)=det(I-BA).)

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