Chapter 5

Question 1

Eigenvector (x) of A if:

Answer 1

Question 2

An (n x n) matrix has how many eigenvalues?

Answer 2

eigenvalues ≤ n

Question 3

λ is an eigenvalue of A iff:

Answer 3

Question 4

How do you find the eigenvalues of a TRIANGULAR matrix?

Answer 4

λ of A are the entries on the main diagonal of A.

Question 5

Similarity invariants: properties that are preserved under similarity transformations

Answer 5

DETERNIC

Determinant, eigenvalues, trace, eigenspace dimension, rank, nulity, invertibility, characteristic polynomial

Question 6

How do you know if B is similar to A?

Answer 6

if there is an invertible matrix P such that B = P^-1AP

Question 7

If B is similar to A then?

Answer 7

If B is similar to A, then A is similar to B

Question 8

What makes a matrix diagonalizable?

Answer 8

if square matrix A is similar to some diagonal matrix

Question 9

If A (n x n) is diagonalizable, then how many linearly independent eigenvectors does it have?

Answer 9

If A is diagonalizable, it has n linearly independent eigenvectors

Question 10

Procedure for diagonalizing an (n x n) matrix:

Answer 10

1. Determine if matrix is diagonalizable (n linearly independent eigenvectors)
2. Form matrix P: column vectors = basis vectors
3. D = P^-1AP

Question 11

Can you determine eigenvalues of A^k if you know the eigenvalues of A?

Answer 11

If λ = eigenvalue of A, then λ^k = eigenvalue of A^k

Question 12

What is geometric multiplicity?

Answer 12

dimension of eigenspace corresponding to λ₀

Question 13

What is algebraic multiciplty?

Answer 13

of times that (λ - λ₀) appears as a factor in the characteristic polynomial of A

Question 14

Example problem:

What is the geometric & algebraic multiplicity of:

(λ - 1)²(λ + 2) for λ = 1

Answer 14

Geometric: ≤ 2

Algebraic = 2

Question 15

What is the relationship between geometric & algebraic multiplicity?

Answer 15

geometric multiplicity ≤ algebraic multiplicity

Question 16

Differential equation

Answer 16

equation involving unknown functions & derivatives

Question 17

Difference between general & particular solution of diff equation?

Answer 17

General solution: in terms of C

Particular solution: given an initial condition → solve for C

Question 18

Constant coefficient first-order homogeneous linear system

Answer 18

y’ = Ay

Question 19

Procedure for solving y’ = Ay:

Answer 19

1. Find eigenvalues: det(λI-A) = 0
2. Find bases for eigenspace: plug λ back in and RREF
3. Construct P (using eigenvectors as columns) & D (using eigenvalues)
4. Double-check using: P^-1AP = D
5. Solve: u’ = Du
6. Solve: y = Pu
7. Plug-in initial conditions & solve particular solution

Question 20

Markov chain

Answer 20

stochastic model where probability of each event depends only on the state attained in the previous event

Question 21

What matrix a stochastic matrix P regular?

Answer 21

If all the entries of P^k are positive

Question 22

What a Markov Chain’s steady state vector?

Answer 22

long-term probability → equilibrium of Markov chain

Question 23

What is the equation to find the steady-state vector?

Answer 23