Eigenvalues and Eigenvectors Practice

Find the eigenvalues A^T and A^-1 of matrix A

• A^T: λ=1,4,7
  A^-1: λ=1,1/4,1/7
• A^T: λ=1,5,9
  A^-1: λ=1,1/5,1/9
• A^T: λ=0,5,8
  A^-1: λ=0,1/5,1/8
• A^T: λ=1,0,0
  A^-1: λ=undefined

Find the eigenvalues for A⁹ of matrix A

• λ=1,1/8,512
• λ=1,1/2,2
• λ=1,1/512,512
• λ=1,0,512

Find the dimension of eigenspace/geometric multiplicity for matrix A.

• 1
• 2
• 3
• 4

Find eigenvalues for matrix A

• λ=1,3
• λ=-1,2
• λ=1,-2
• λ=1,4

If A is an nxn matrix, then the eigenvalues can be determined via ?

The eigenvalues of matrix A are ...

• A has no eigenvalues

How are eigenvectors related to eigenvalues?

• Eigenvectors are related to eigenvalues as the vectors associated with specific eigenvalues.

• Eigenvalues are the reciprocals of eigenvectors

• Eigenvectors have no relationship to eigenvalues

• Eigenvectors are always orthogonal to eigenvalues

Discuss the importance of eigenvalues and eigenvectors in machine learning.

• Eigenvalues and eigenvectors are used for data encryption in machine learning.

• Eigenvalues and eigenvectors are important in machine learning for dimensionality reduction, feature extraction, and understanding data structure.

• Eigenvalues and eigenvectors are only relevant in theoretical aspects of machine learning.

• Eigenvalues and eigenvectors have no impact on the accuracy of machine learning models.

What is the rank of a matrix?

• The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix.

• The rank of a matrix is the sum of all elements in the matrix.

• The rank of a matrix is the determinant of the matrix.

• The rank of a matrix is the product of the elements on the main diagonal of the matrix.

Find the Eigenvalues of the matrix E = [3 1; 2 4].

• The eigenvalues of the matrix E = [3 1; 2 4] are λ1 = 2 and λ2 = 5.

• λ1 = 0 and λ2 = 7

• λ1 = 1 and λ2 = 6

• λ1 = 3 and λ2 = 4