6.3. EXERCISES 157

53 − 1

3 − 13

− 13

76

16

− 13

16

76

 The eigenvalues are 1, 2, and 1. What is the physical interpreta-

tion of the repeated eigenvalue?

35. Find oscillatory solutions to the system of differential equations, x′′ = Ax where A = −3 −1 −1

−1 −2 0

−1 0 −2

 The eigenvalues are −1,−4, and −2.

36. Let A and B be n× n matrices and let the columns of B be

b1, · · · ,bn

and the rows of A areaT1 , · · · ,aTn .

Show the columns of AB areAb1 · · ·Abn

and the rows of AB areaT1 B · · ·aTnB.

37. 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 i

1 + 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.

38. Let M be an n × n matrix and suppose x1, · · · ,xn are n eigenvectors which form alinearly independent set. Form the matrix S by making the columns these vectors.Show that S−1 exists and that S−1MS is a diagonal matrix (one having zeros every-where except on the main diagonal) having the eigenvalues ofM on the main diagonal.When this can be done the matrix is said to be diagonalizable.

39. Show that a n×n matrix M is diagonalizable if and only if Fn has a basis of eigenvec-tors. Hint: The first part is done in Problem 38. It only remains to show that if thematrix can be diagonalized by some matrix S giving D = S−1MS for D a diagonalmatrix, then it has a basis of eigenvectors. Try using the columns of the matrix S.

40. Let

A =

1 2

3 4

2

0

0 1 3

and let

B =

0 1

1 1

2 1

