Quiz 6

Question 1

What is an eigenvector of a matrix A?

Answer 1

An eigenvector is a vector v such that there exists some scalar x for which
Av = xv

Question 2

What is the characteristic equation of A and how can it be used?

Answer 2

det(A-λI) = 0
It was be used to solve for λ, which gives us the eigenvalues of a matrix

Question 3

What do you need to review from last chapter?

Answer 3

FINDING DETERMINANTS

Question 4

How do you find the eigenvector corresponding to an eigenvalue?

Answer 4

Use the equation
(A-λI)x = 0
Plug in your found eigenvalue and solve for the vector x

Question 5

What is true of similar matrices?

Answer 5

They have the same characteristic polynomial (and as a result the same eigenvalues)

If two matrices do not have the same characteristic equation they are not similar

Question 6

What is the eigenspace?

Answer 6

An eigenspace is a vector space specific to a given λ of a given matrix.

It is the kernel of (A-λI)

Essentially it is the space containing all eigenvectors for a given eigenvalue

Question 7

What problem do you need to figure out in chapter 4.1?

Answer 7

Example 7

Question 8

What is the sum of all a matrix’s eigenvalues?

Answer 8

Their sum is equal to the trace of their matrix

Question 9

What is true of an upper or lower triangular and diagonal matrix?

Answer 9

Their eigenvalues are the elements of the main diagonal

Question 10

What is a singular matrix and what is true in relation to eigenvalues?

Answer 10

A matrix is singular if and only if it has a zero eigenvalue

Question 11

What is true of the inverse of an invertible matrix A?

Answer 11

If x is an eigenvector in A according to eigenvalue λ, then it is also an eigenvector in A^-1 corresponding to 1/λ

Question 12

What are some equivalencies to all eigenvalues of A being non-zero?

Answer 12

A has an inverse. A has rank n. A can be transformed to an upper diagonal using elementary row operations. A has a non-zero determinant

Question 13

What is true for a matrix kA?

Answer 13

If x is an eigenvector of A for eigenvalue λ then x is an eigenvector for kA corresponding to eigenvalue kλ

Question 14

What is the product of all eigenvalues of a matrix?

Answer 14

It’s determinant

Question 15

What is true for a matrix A^n?

Answer 15

If x is an eigenvector of A for eigenvalue λ then x is an eigenvector for A^n corresponding to eigenvalue λ^n

Question 16

What is true of a matrix A - cI?

Answer 16

If is an eigenvector of A corresponding to eigenvalue λ then for any scalar c, λ-c and x are a corresponding pair of eigenvectors and eigenvalues in A - cI

Question 17

What is important to remember when working with basis transformations for finding eigenvalues?

Answer 17

The final eigenvectors which you have for the eigenspaces must be replugged into the original basis equation to get the basis with respect to the proper basis