4 FALL 2017 Theorem 5.If B=S-1AS,then .pA(t)=pB(t),i.e.A and B have the same eigenvalues with the same algebraic multiplicities. Theorem 6.For AM(C),the following are equivalent: (a)A is diagonalisable (b)A has n linearly independent eigenvectors; (c)for A E o(A).its algebraic multiplicity equals to its geometric multiplicity. Theorem 7.Suppose C=where AEM(C)and BEMm(C).Then C is diagonalisable if and 3.MORE ABOUT EIGENVECTOR Definition 3.For AEMn(C), 。ifAr=入r for some x≠0，then r is called the right eigenvector associated with入. ·fyA=Ay'for somey≠0，then y is called the left eigenvector associated with入 Theorem9.Suppose is a right eigenvector of A associated with andy is a left eigenvector of A associated ith4.f入≠h,then y'x=0.4 FALL 2017 Theorem 5. If B = S −1AS, then • pA(t) = pB(t), i.e. A and B have the same eigenvalues with the same algebraic multiplicities. • If Ax = λx (x 6= 0), then letting y = S −1x, n 6= 0 and By = λy. So A and B have the same eigenvalues with the same geometric multiplicities. Theorem 6. For A ∈ Mn(C), the following are equivalent: (a) A is diagonalisable; (b) A has n linearly independent eigenvectors; (c) for λ ∈ σ(A), its algebraic multiplicity equals to its geometric multiplicity. Corollary 1. For A ∈ Mn(C), if A has n distinct eigenvalues, then A is diagonalisable. (The reverse is not true.) Theorem 7. Suppose C =  A 0 0 B  where A ∈ Mn(C) and B ∈ Mm(C). Then C is diagonalisable if and only if A and B are both diagonalisable. 3. More about eigenvector Definition 3. For A ∈ Mn(C), • if Ax = λx for some x 6= 0, then x is called the right eigenvector associated with λ. • if y ∗A = λy∗ for some y 6= 0, then y is called the left eigenvector associated with λ. Theorem 8. If λ1, ..., λk are k distinct eigenvalues of A and αi is an eigenvector associated with λi. Then {α1, ..., αk} is a set of linearly independent eigenvectors. Theorem 9. Suppose x is a right eigenvector of A associated with λ and y is a left eigenvector of A associated with µ. If λ 6= µ, then y ∗x = 0