Problem 1
(a). Let and Each Be Matrices with Distinct Eigenvalues. Suppose and Have the Same Eigenvectors. Prove that
(b). Find Matrices and such that is a Basis for the 1-eigenspace of and the 2-eigenspace of , and is a Basis for the 3-eigenspace of and the 4-eigenspace of then Verify that by Calculating the Product
Problem 2
(a). Let and Verify that and Each Have 2 Distinct Eigenvalues, and that
(b). Let and Each Be Matrices with Distinct Eigenvalues, and Suppose . Prove that and Have the Same Eigenvectors
is an eigenvector of with eigenvalue since has distinct eigenvalues this means that each eigenspace has only 1 dimension, it therefore follows that if is in the eigenspace it must be some multiple of via the definition of the eigenvector this shows that for some constant meaning both and have the same eigenvector .
(c). Let and Be as in part (a) Verify that and Have the Same Eigenvectors by Finding Bases for Their Eigenspaces
Problem 3
(a). Come up with a Diagonalizable Matrix that Has Characteristic Polynomial . the Row Reduce the Matrix
(b). In General, Let Be a Diagonalizable Matrix in with Characteristic Polynomial how Many Pivots Does Have?
The matrix will have pivots.
Problem 4
Let be a standard matrix of the linear transformation that reflects vectors across the line .
(a). Has Two Eigenvalues. Find Them, and for Each, Write down the Basis of Their Eigenspace
(b). Find , Making Sure to Simplify All the Entries
Via the rotation-scaling theorem we know that the transformation from to will scale by which is just 1 in this case, where therefore know that and our final solution is