Eigenvalues & Eigenvectors

Question 1

What is the formula for eigenvalues & eigenvectors?

Answer 1

Ax = λx

Question 2

What does x represent in the eigenvalue formula?

Answer 2

The eigenvector, a vector with the same length as A.

Question 3

What does A represent in the eigenvalue formula?

Answer 3

A square matrix of known value

Question 4

What does lambda represent in the eigenvalue formula?

Answer 4

The eigenvalue, a scaler

Question 5

How do you find the Eigenvalues?

Answer 5

Subtract lambda from the main diagonal of A, set the determinant equal to 0, & solve for lambda

Question 6

How many eigenvalues & eigenvectors are there for a given square matrix A?

Answer 6

The number of each will be less then or equal to the length of A. If A is 10x10, there will be up to 10 eigenvalues & the same number of eigenvectors.

Question 7

Practice finding the eigenvalues & eigenvectors for a 3x3 matrix

Answer 7

Check using the formula.

Question 8

What does tr(A) mean?

Answer 8

The trace of matrix A.

Question 9

What is the trace of a matrix?

Answer 9

The sum of the main diagonal.

Question 10

What is the relationship between the trace & the eigenvalues?

Answer 10

The trace is equal to the sum of eigenvalues. Beware of multiplicities.