Exam 6: Eigenvalues and Eigenvectors

Find the matrix A that has the given eigenvalues and bases for the corresponding eigenspaces.

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

Factor the given matrix A in the form where B is a rotation-dilation matrix. Find the dilation and angle of rotation. Use this information to evaluate the matrix power without computing it directly.

Find the rotation-dilation matrix B within the given matrix A.

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

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.

Determine the rotation and dilation that result from multiplying vectors in by the given matrix.

Diagonalize the matrix A, if possible.

Determine which of , , and are eigenvectors of , and determine the associated eigenvalues.

If and , where , and and are nonzero vectors, then is linearly independent.

Compute if .

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

Factor the matrix from Question 2 in the form where B is a rotation-dilation matrix.

If A and are diagonalizable matrices, then AB is diagonalizable.

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.

Find the general solution for the system .

If an matrix A has n distinct eigenvalues, then A is diagonalizable.

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

Find the matrix A that has the given eigenvalues and bases for the corresponding eigenspaces. ; ;

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