Questions: Concept 11-11-2011
MA2940: Linear Algebra for Engineers
Diagonalization
1) (T/F) If = 0 is an eigenvalue of a square matrix A, then A is not diagonalizable.
2) (T/F) If 16 is an eigenvalue of A4 , then 2 is an eigenvalue of A.
3) (T/F) Suppose n n matrices A and B can be diagonalized with the same eigenvector
matrix S . Then AB = BA.
4) (T/F) Suppose n n matrices A and B can be diagonalized, and AB = BA. Then they
can be diagonalized with the same matrix S of eigenvectors.
Applications to Dierential Equations
5) (T/F) It is possible to have eigenvalues = 1 2 that share the same eigenvector.
6) (T/F) It is possible to have eigenvectors v1 = v2 that share the same eigenvalue.
7) (T/F) Any matrix system
tial equation.
d
u(t)
dt
= Au(t) an be written as a single higher-order dieren-
8) (T/F) If w(t) and v (t) are solutions to
d
u(t)
dt
= Au(t) + b, then w + v is a solution too.
9) (T/F) If a square matrix A is unstable, then tr(A) > 0 and det(A) < 0.
d
10) When solving dt u(t) = Au(t), does it matter if A isnt diagonalizable? (Yes.) Why
doesnt our method work? What do we do?
TA: Kyle Wilson