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1. Consider the matrices 10 A₁ = -( : ) - ( ) - ( 1 ) 020 A3 3 3 A₂ 02 00 -1 0 0-2 1 Compute the eigenvalues and eigenvectors of each A..

1. Consider the matrices 10 A₁ = -( : ) - ( ) - ( 1 ) 020 A3 3 3 A₂ 02 00 -1 0 0-2 1 Compute the eigenvalues and eigenvectors of each A..

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section5.5: Applications
Problem 74EQ
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4:24 1. Consider the matrices. mtaylor.web.unc.edu 1 0 1 2 A₁ 02 - (₁ ¦ ¦) + - ( ¦ ¦) ^ - ( ¹ ) 0 A₂ = 020, A33 1 -1 0 -1 0-2 Exercises Compute the eigenvalues and eigenvectors of each Aj. 2. Find the minimal polynomial of A; and find a basis of generalized eigen- vectors of Aj. 3. Consider the transformation D: P2 P2 given by (1.4.15). Find the eigenvalues and eigenvectors of D. Find the minimal polynomial of D and find a basis of P2 consisting of generalized eigenvectors of D. 4. Suppose V is a finite dimensional complex vector space and T: V→V. Show that V has a basis of eigenvectors of T if and only if all the roots of the minimal polynomial mr(A) are simple. 5. In the setting of (2.2.3)-(2.2.4), given SE L(V), show that ST=TS⇒S: GE(T, Aj) →GE(T, Aj). Exercises 6. Show that if V is an n-dimensional complex vector space, S, T = L(V), and ST TS, then V has a basis consisting of vectors that are simultane- ously generalized eigenvectors of T and of S. Hint. Apply Proposition 2.2.6 to S: GE(T, X₁)→GE(T, A₁). 7. Let V be a complex n-dimensional vector space, and take TE L(V), with minimal polynomial mȚ(A), as in (2.2.13). For l {1,..., K}, set mT(X) P(X) = x-de Show that, for each le {1,..., K}, there exists we V such that ve = Pe(T) we 0. Then show that (TAI)ve = 0, so one has a proof of Proposition 2.1.1 that does not use determinants. 71 8. In the setting of Exercise 7, show that the exponent k; in (2.2.14) is the smallest integer such that (TAI) annihilates GE (T, Aj). Hint. Review the proof of roposition 2.2.4.
Transcribed Image Text:4:24 1. Consider the matrices. mtaylor.web.unc.edu 1 0 1 2 A₁ 02 - (₁ ¦ ¦) + - ( ¦ ¦) ^ - ( ¹ ) 0 A₂ = 020, A33 1 -1 0 -1 0-2 Exercises Compute the eigenvalues and eigenvectors of each Aj. 2. Find the minimal polynomial of A; and find a basis of generalized eigen- vectors of Aj. 3. Consider the transformation D: P2 P2 given by (1.4.15). Find the eigenvalues and eigenvectors of D. Find the minimal polynomial of D and find a basis of P2 consisting of generalized eigenvectors of D. 4. Suppose V is a finite dimensional complex vector space and T: V→V. Show that V has a basis of eigenvectors of T if and only if all the roots of the minimal polynomial mr(A) are simple. 5. In the setting of (2.2.3)-(2.2.4), given SE L(V), show that ST=TS⇒S: GE(T, Aj) →GE(T, Aj). Exercises 6. Show that if V is an n-dimensional complex vector space, S, T = L(V), and ST TS, then V has a basis consisting of vectors that are simultane- ously generalized eigenvectors of T and of S. Hint. Apply Proposition 2.2.6 to S: GE(T, X₁)→GE(T, A₁). 7. Let V be a complex n-dimensional vector space, and take TE L(V), with minimal polynomial mȚ(A), as in (2.2.13). For l {1,..., K}, set mT(X) P(X) = x-de Show that, for each le {1,..., K}, there exists we V such that ve = Pe(T) we 0. Then show that (TAI)ve = 0, so one has a proof of Proposition 2.1.1 that does not use determinants. 71 8. In the setting of Exercise 7, show that the exponent k; in (2.2.14) is the smallest integer such that (TAI) annihilates GE (T, Aj). Hint. Review the proof of roposition 2.2.4.
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