Linear Algebra

Section II. Similarity 3593.23 Find the characteristic polynomial, the eigenvalues, and the associated eigenvectorsof this matrix. ( 1 1) 10 0 10 0 1̌ 3.24 For each matrix, find the characteristic equation, and the eigenvalues andassociated eigenvectors.( ) ( )3 −2 00 1 0(a) −2 3 0 (b) 0 0 10 0 54 −17 8̌ 3.25 Let t: P 2 → P 2 bea 0 + a 1x + a 2x 2 ↦→ (5a 0 + 6a 1 + 2a 2) − (a 1 + 8a 2)x + (a 0 − 2a 2)x 2 .Find its eigenvalues and the associated eigenvectors.3.26 Find the eigenvalues and eigenvectors of this map t: M 2 → M 2 .( ) ( )a b 2c a + c↦→c d b − 2c ď 3.27 Find the eigenvalues and associated eigenvectors of the differentiation operatord/dx: P 3 → P 3.3.28 Prove that the eigenvalues of a triangular matrix (upper or lower triangular)are the entries on the diagonal.̌ 3.29 Find the formula for the characteristic polynomial of a 2×2 matrix.3.30 Prove that the characteristic polynomial of a transformation is well-defined.̌ 3.31 (a) Can any non-⃗0 vector in any nontrivial vector space be a eigenvector?That is, given a ⃗v ≠ ⃗0 from a nontrivial V , is there a transformation t: V → Vand a scalar λ ∈ R such that t(⃗v) = λ⃗v?(b) Given a scalar λ, can any non-⃗0 vector in any nontrivial vector space be aneigenvector associated with the eigenvalue λ?̌ 3.32 Suppose that t: V → V and T = Rep B,B (t). Prove that the eigenvectors of Tassociated with λ are the non-⃗0 vectors in the kernel of the map represented (withrespect to the same bases) by T − λI.3.33 Prove that if a, . . . , d are all integers and a + b = c + d then( )a bchas integral eigenvalues, namely a + b and a − c.̌ 3.34 Prove that if T is nonsingular and has eigenvalues λ 1 , . . . , λ n then T −1 haseigenvalues 1/λ 1, . . . , 1/λ n. Is the converse true?̌ 3.35 Suppose that T is n×n and c, d are scalars.(a) Prove that if T has the eigenvalue λ with an associated eigenvector ⃗v then ⃗vis an eigenvector of cT + dI associated with eigenvalue cλ + d.(b) Prove that if T is diagonalizable then so is cT + dI.̌ 3.36 Show that λ is an eigenvalue of T if and only if the map represented by T −λIis not an isomorphism.3.37 [Strang 80](a) Show that if λ is an eigenvalue of A then λ k is an eigenvalue of A k .(b) What is wrong with this proof generalizing that? “If λ is an eigenvalue of Aand µ is an eigenvalue for B, then λµ is an eigenvalue for AB, for, if A⃗x = λ⃗xand B⃗x = µ⃗x then AB⃗x = Aµ⃗x = µA⃗xµλ⃗x”?d