Eigenvalues and Matrix Powers

The video tutorial explains how to use eigenvalues and eigenvectors to simplify the calculation of matrix powers. It begins by introducing the concept of eigenvalues and eigenvectors, then demonstrates how to calculate matrix powers using these concepts. Two examples are provided: one with an eigenvalue of -1 and eigenvector u, and another with an eigenvalue of 2 and eigenvector v. The tutorial concludes with a summary of the calculations and their results.

What are the eigenvalues of matrix A?

• Negative one and three

• Negative one and two

• Two and three

• One and two

Which vectors correspond to the eigenvalues of matrix A?

• Vectors u and w

• Vectors v and w

• Vectors u and v

• Vectors x and y

What is the formula for calculating matrix A to the power of n times an eigenvector?

• A^n * v = n * lambda * v

• A^n * v = lambda * v^n

• A^n * v = lambda^n * v

• A^n * v = lambda^n * A

What is the result of matrix A to the power of seven times eigenvector u?

• Vector one five

• Vector negative one five

• Vector five negative one

• Vector negative five one

What is the seventh power of the eigenvalue corresponding to eigenvector u?

• One

• Negative one

• Zero

• Two

What is the result of matrix A to the power of seven times eigenvector v?

• Vector 128 256

• Vector 640 512

• Vector 256 128

• Vector 512 640

What is the seventh power of the eigenvalue corresponding to eigenvector v?

• 64

• 256

• 128

• 512

How is the vector 640 512 obtained in the calculation?

• By multiplying 256 with vector 4 5

• By multiplying 128 with vector 4 5

• By multiplying 256 with vector 5 4

• By multiplying 128 with vector 5 4

What is the significance of eigenvalues in computing matrix powers?

• They are irrelevant

• They simplify the computation

• They complicate the computation

• They are only used for 2x2 matrices

What is the final result of matrix A to the power of seven times eigenvector v?

• Vector 256 128

• Vector 512 640

• Vector 640 512

• Vector 128 256