Exam 6: Eigenvalues and Eigenvectors

Compute the first three iterations of the Power Method without scaling, starting with the given , where .

Find the characteristic polynomial, the eigenvalues, and a basis for each eigenspace for the matrix A = .

Find the solution for the system that satisfies the condition at t = 0.​

If A is a real square matrix with complex eigenvalue and associated eigenvector , then is a real solution to the system .

If A is a real matrix and is an eigenvalue of A with and corresponding eigenvector u, then .

The dominant eigenvalue of the matrix A given below is . Compute the first two iterations of the Shifted Power Method with scaling, starting with the given , to determine the eigenvalue farthest from , rounding any numerical values to two decimal places.

Find the characteristic polynomial, the eigenvalues, and a basis for each eigenspace for the matrix A = .

The coefficient matrix for a system of linear differential equations of the form has the given eigenvalues and eigenspace bases. Find the general solution for the system.

If u and v are both eigenvectors of an n ×n matrix A, then u+v is also an eigenvector of the A.

Compute the first two iterations of the Inverse Power Method, starting with the given , rounding any numerical values to two decimal places.

Find the general solution for the system .

If A is a real matrix, and is a complex eigenvalue of A, then is also an eigenvalue of A.

An matrix A can have no more than n eigenvalues.

Use the Power Method with scaling to determine an eigenvalue and associated eigenvector of A, starting with the given .

Find the eigenvalues and a basis for each eigenspace for the given matrix.