finals: sections 5.1-5.8

Question 1

what is an eigenvector of an nxn matrix A?

Answer 1

nonzero vector x such that Ax = λx

Question 2

when is a scalar λ called an eigenvalue?

Answer 2

if there is a nontrivial solution x of
Ax=λx

Question 3

what are the eigenvalues of a triangular matrix?

Answer 3

the entries on its diagonal

Question 4

what is the eigenspace of an nxn matrix A corresponding to λ?

Answer 4

it is the nullspace of the matrix A - λI. that is, the eigenspace consists of the zero vector and all eigenvectors corresponding to λ

Question 5

if zero is NOT an eigenvalue for a matrix A _____________

Answer 5

then A is invertible

Question 6

a scalar λ is an eigenvalue of an nxn matrix A if and only if it satisfies the characteristic equation __________

Answer 6

det(A - λI) = 0

Question 7

an nxn matrix is diagonalizable if and only if ___________________

Answer 7

it has n linearly independent eigenvectors

Question 8

when does finding the eigenvalues of a matrix become diffucult?

Answer 8

when the matrix is large,
has complex entries,
or the characteristic polynomial is difficult to factor

Question 9

what is the power method used for?

Answer 9

it is used to estimate the strictly dominant eigenvalue

Question 10

what is the strictly dominant eigenvalue?

Answer 10

λ is strictly GREATER THAN all the other eigenvalues of the matrix

Question 11

what are the steps for the power method?

Answer 11

1) select an initial vector x0, whose largest entry is 1.
2) For k = 0; 1; …
a) Compute Axk
b) Let μk be an entry in Axk whose absolute value is as large as possible.
c) Compute xk+1 = (1/μk) * Axk

Question 12

what is the dominant eigenvalue in the power method?

Answer 12

μk

Question 13

what is the dominant eigenvector in the power method?

Answer 13

xk