Indian Institute of Science Education and Research Assignment 4 MTH 320 - Vector Space, Rings and Modules Dimension of vector space is n. Exercise 1. Let T ∈ End(V ). Further suppose V = V1 ⊕ V2 ⊕ · · · ⊕ Vk such that T (Vi ) ⊂ Vi . Denote Ti = T |Vi , the restriction of T to Vi . Then, (1) Show that we can choose a basis of V such that the matrix of T is block diagonal. (2) Let mi (X) be the minimal polynomial of Ti . What is the minimal polynomial of T ? (3) Let χi (X) be the characteristic polynomial of Ti . What is χT (X)? Exercise 2. Let Jr (λ) be the r × r matrix with all diagonals λ and 1 at supper diagonals (above diagonal). (1) Find minimal and characteristic polynomial of Jr (λ)? (2) Find minimal and characteristic polynomial of Jr¯(λ) := Jr1 (λ) ⊕ Jr2 (λ) ⊕ · · · ⊕ Jrk (λ) where notation r¯ = (r1 , . . . , rk ) written in decreasing order is a partition. (3) Find minimal and characteristic polynomial of J := Jn1 (λ1 ) ⊕ Jn2 (λ2 ) ⊕ · · · ⊕ Jnm (λm ). (4) Find minimal and characteristic polynomial of J := Jn¯ 1 (λ1 ) ⊕ Jn¯ 2 (λ2 ) ⊕ · · · ⊕ Jn¯ m (λm ). Exercise 3. Let A be a diagonal matrix. Suppose B is another diagonal matrix with same entries on the diagonal except that they may be permuted. Show by direct computation that they are similar. Exercise 4. Let T be a linear transformation such that T 2 = 1. Show that V = V1 ⊕ V2 as I T -invariant subspaces such that T can be conjugated to a matrix p . In−p Exercise 5. Let T be a projection, i.e., T 2 = T . Is T diagonalisable? Exercise 6. Let g ∈ GLn (C) be of finite order, i.e., g r = 1 for some r. Is g diagonalisable? Exercise 7. Write down canonical forms of all nilpotent as well as unipotent matrices up to n ≤ 6 and its invariants. Exercise 8. (1) Let T ∈ GL2 (C). Write down representatives of each conjugacy classes. (2) Let T ∈ GL2 (R). Write down representatives of each conjugacy classes. Exercise 9. Let k = C. Show that any matrix is similar to its transpose. Exercise 10. How does a matrix change if we conjugate it by (A) a diagonal matrix, (B) a permutation matrix, or (C) matrices of the form I + teij where i 6= j. Use these operations to get Jordan-canonical form of any 3 × 3 and 4 × 4 upper triangular matrices.

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