Exam 6: Eigenvalues and Eigenvectors

Diagonalize the matrix A, if possible.

Find a basis for the eigenspace associated with eigenvalue for matrix .

Diagonalize the given matrix A, and use the diagonalization to compute .

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.

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

Suppose that A is an matrix and is a solution to the system of linear differential equations where is an eigenvector of A with associated eigenvalue . If A is invertible, then is a solution to the system .

Suppose is a matrix having eigenvalues . Then has dominant eigenvalue .

If , and are solutions to the initial-value problem , with , then for all t.

If the invertible matrix A has hidden rotation-dilation matrix , where then has hidden rotation-dilation matrix .

The Shifted Inverse Power Method for an invertible n×nmatrix Ais implemented by applying the Power Method to for some scalar c.

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

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

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

Find a basis for the eigenspace associated with eigenvalue for the matrix .

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

Diagonalize the given matrix A, and use the diagonalization to compute .

Compute if .

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

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

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