158 CHAPTER 6. SPECTRAL THEORY

MultiplyAB verifying the block multiplication formula. HereA11 =

(1 2

3 4

), A12 =(

2

0

), A21 =

(0 1

)and A22 = (3) .

41. Suppose A,B are n×n matrices and λ is a nonzero eigenvalue of AB. Show that thenit is also an eigenvalue of BA. Hint: Use the definition of what it means for λ to bean eigenvalue. That is,

ABx = λx

where x ̸= 0. Maybe you should multiply both sides by B.

42. Using the above problem show that if A,B are n× n matrices, it is not possible thatAB − BA = aI for any a ̸= 0. Hint: First show that if A is a matrix, then theeigenvalues of A− aI are λ− a where λ is an eigenvalue of A.

43. Consider the following matrix.

C =

0 · · · 0 −a01 0 −a1

. . .. . .

...

0 1 −an−1

Show det (λI − C) = a0+λa1+ · · · an−1λ

n−1+λn. This matrix is called a companionmatrix for the given polynomial.

44. A discreet dynamical system is a relation of the following form in which x(k) is a n×1vector and A is a n× n square matrix.

x (k + 1) = Ax (k) , x (0) = x0

Show first thatx (k) = Akx0

for all k ≥ 1. If A is nondefective so that it has a basis of eigenvectors, {v1, · · · ,vn}where

Avj = λjvj

you can write the initial condition x0 in a unique way as a linear combination of theseeigenvectors. Thus

x0 =

n∑j=1

ajvj

Now explain why

x (k) =

n∑j=1

ajAkvj =

n∑j=1

ajλkjvj

which gives a formula for x (k) , the solution of the dynamical system.

45. Suppose A is an n × n matrix and let v be an eigenvector such that Av = λv. Alsosuppose the characteristic polynomial of A is

det (λI −A) = λn + an−1λn−1 + · · ·+ a1λ+ a0