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2. Let n>1 and let T: M₁(k) →→→ M₁(k) be the linear map defined by - T(A) = A + A¹ For k= R, C, F₂ determine the minimal polynomial of T (justify your answer) and in each case state whether T is diagonalisable. What are the eigenvalues of T? (Hint. Start by looking at T2...) Suppose k R. Find the characteristic polynomial chr. = (you may use without proof that the dimension of the space of symmetric matrices is n(n+1) and that of antisymmetric matrices is n(n-1)) 2 2

2. Let n>1 and let T: M₁(k) →→→ M₁(k) be the linear map defined by - T(A) = A + A¹ For k= R, C, F₂ determine the minimal polynomial of T (justify your answer) and in each case state whether T is diagonalisable. What are the eigenvalues of T? (Hint. Start by looking at T2...) Suppose k R. Find the characteristic polynomial chr. = (you may use without proof that the dimension of the space of symmetric matrices is n(n+1) and that of antisymmetric matrices is n(n-1)) 2 2

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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2. Let n 1 and let T: M₁(k) → M₁(k) be the linear map defined by T(A) = A + At For k = R, C, F₂ determine the minimal polynomial of T (justify your answer) and in each case state whether T is diagonalisable. What are the eigenvalues of T? (Hint. Start by looking at T2...) Suppose k R. Find the characteristic polynomial chr. (you may use without proof that the dimension of the space of symmetric matrices is n(n+1) and that of antisymmetric matrices is n(n-1)) 2
Transcribed Image Text:2. Let n 1 and let T: M₁(k) → M₁(k) be the linear map defined by T(A) = A + At For k = R, C, F₂ determine the minimal polynomial of T (justify your answer) and in each case state whether T is diagonalisable. What are the eigenvalues of T? (Hint. Start by looking at T2...) Suppose k R. Find the characteristic polynomial chr. (you may use without proof that the dimension of the space of symmetric matrices is n(n+1) and that of antisymmetric matrices is n(n-1)) 2
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