12.5. EXERCISES 285

43. Is it possible for a nonzero matrix to have only 0 as an eigenvalue?

44. Let M be an n× n matrix. Then define the adjoint of M,denoted by M∗ to be thetranspose of the conjugate of M. For example,(

2 i1+ i 3

)∗=

(2 1− i−i 3

).

A matrix M, is self adjoint if M∗ = M. Show the eigenvalues of a self adjoint matrixare all real. If the self adjoint matrix has all real entries, it is called symmetric.

45. Suppose A is an n×n matrix consisting entirely of real entries but a+ ib is a complexeigenvalue having the eigenvector x+ iy. Here x and y are real vectors. Show thatthen a− ib is also an eigenvalue with the eigenvector x− iy. Hint: You shouldremember that the conjugate of a product of complex numbers equals the product ofthe conjugates. Here a+ ib is a complex number whose conjugate equals a− ib.

46. Recall an n×n matrix is said to be symmetric if it has all real entries and if A = AT .Show the eigenvectors and eigenvalues of a real symmetric matrix are real.

47. Recall an n× n matrix is said to be skew symmetric if it has all real entries and ifA =−AT . Show that any nonzero eigenvalues must be of the form ib where i2 =−1.In words, the eigenvalues are either 0 or pure imaginary.

48. A discreet dynamical system is of the form

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

where A is an n×n matrix and x(k) is a vector in Rn. Show first that

x(k) = Akx0

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

Av j = λ jv j

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

x0 =n

∑j=1

a jv j

Now explain why

x(k) =n

∑j=1

a jAkv j =n

∑j=1

a jλkjv j

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

49. 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(λ I−A) = λn +an−1λ

n−1 + · · ·+a1λ +a0