Module 6 Quiz: Eigenvalues and Eigenvectors
Quiz
10 questions on eigenvalues, eigenvectors, diagonalization, and applications.
1
What equation defines an eigenvector v with eigenvalue lambda?
Av = lambda v, where v is nonzero.
2
Find the eigenvalues of A = [7 -2; 0 3].
Triangular matrix. Eigenvalues: 7 and 3.
3
What is the characteristic polynomial of A = [1 3; 4 2]?
p(lambda) = lambda^2 - 3*lambda + (2-12) = lambda^2 - 3*lambda - 10 = (lambda-5)(lambda+2).
4
True or False: The zero vector can be an eigenvector.
False. By definition, eigenvectors must be nonzero.
5
A has eigenvalue lambda = 0. What does this tell you about A?
A is singular (not invertible). det(A) = 0 since the product of eigenvalues includes 0.
6
When is an n x n matrix guaranteed to be diagonalizable?
When it has n distinct eigenvalues, or more generally when geometric multiplicity equals algebraic multiplicity for every eigenvalue.
7
If A = PDP^{-1}, express A^n in terms of P and D.
A^n = PD^nP^{-1}. D^n is obtained by raising each diagonal entry to the nth power.
8
What is a steady-state vector in the context of Markov chains?
A probability vector q (nonnegative entries summing to 1) such that Aq = q. It is an eigenvector for eigenvalue 1.
9
Matrix A has eigenvalues 1.2 and 0.8. Does the system x_{k+1} = Ax_k converge?
No. |1.2| > 1, so the component along the eigenvector for 1.2 grows without bound.
10
A 3x3 matrix has eigenvalues 2, 2, 5. The eigenspace for lambda=2 has dimension 2. Is A diagonalizable?
Yes. Geometric mult of lambda=2 is 2 = algebraic mult. Geometric mult of lambda=5 is 1. Total independent eigenvectors: 2+1=3=n.